**General topology**, **point-set topology**, or **analytic toplogy** is the study of
the invariant properties of objects under continuous transformations, e.g. stretch, twist, scaling.
In comparison, geometry is constrained to isometries, e.g. translation, rotation, bending.
Topology can be further divided into differential topology, algebraic topology
(including combinatorial topology), low-dimensional topology ($\dim M = 2, 3, 4$), etc.

Open **topology** or **topological structure** $(\mathcal{T}, (\cup_\alpha, \cap))$
is an algebraic system consisting of a subset $\mathcal{T}$ of a power set,
which includes the empty set and the underlying set,
i.e. $\{\emptyset, X\} \subset \mathcal{T} \subset \mathcal{P}(X)$,
and two operations: (1) arbitrary union $\cup_\alpha$ and (2) intersection $\cap$
(such that the system is closed under arbitrary union and finite intersection).
**Closed topology** $(\mathcal{T}^∗, (\cap_\alpha, \cup))$
is the dual concept of open topology, which exchanges union with intersection.
Given a topological structure $(\mathcal{T}, (∗_\alpha, \star))$ of either type for a set,
the other $(\mathcal{T}^∗, (\star_\alpha, ∗))$ is its dual consisting of the complement elements,
i.e. complementation $\complement: \mathcal{T} \mapsto \mathcal{T}^∗$
is an isomorphism between open and closed topologies.
Compare topological structure with sigma-algebra
$(\Sigma, (\cup_{\mathbb{N}}, \cap_{\mathbb{N}}; \complement))$.
A topology is said to be **weaker** than another topology,
and similarly the latter **stronger** than the former,
if the former specifies a coarser topological structure of the underlying set:
$\mathcal{T}_1 \subset \mathcal{T}_2$.
For a set, the weakest topology $\{\emptyset, X\}$ consists of the empty set and itself;
the strongest topology is its power set $\mathcal{P}(X)$, aka the **discrete topology**.

Topological **basis** $\mathcal{B}$ for some topology on a set $X$
is a cover of the set that is "pre-closed" in intersection:
$\bigcup \mathcal{B} = X$; $\forall x \in B_1 \cap B_2$, $B_1, B_2 \in \mathcal{B}$,
$\exists B_3 \in \mathcal{B}$: $x \in B_3 \subset B_1 \cap B_2$.
**Topology generated by a basis** $\mathcal{B}$ is the topology $\mathcal{T(B)}$
consisting of unions of members of arbitrary subsets of the basis:
$\mathcal{T(B)} := \{\bigcup \mathcal{C} \mid \mathcal{C} \subset \mathcal{B}\}$.
If a class of subsets of a set generates a topology on the set,
it is a basis of a topology on the set.
The topology generated by a basis is the weakest topology containing the basis:
$\mathcal{T(B)} = \cap_{\mathcal{B} \subset \mathcal{T}} \mathcal{T}$.
A topology can have many bases, the largest of which is itself.

**Topological space** $(X, \mathcal{T})$ is a set $X$ with a topology $\mathcal{T}$.
**Open subset** $A$ of a topological space, or **open set** in short,
is any element of the topology: $A \in \mathcal{T}$.
**Closed subset** $A$ of a topological space, or **closed set**,
is any element of the closed topology, i.e. its complement is an open subset:
$A \in \mathcal{T}^∗ = \{X \setminus B \mid B \in \mathcal{T}\}$.
For any set endowed with any topological structure,
the empty set and the set itself are both open and closed subsets.

**Closure** (闭包) $\bar{A}$ or $\text{cl}(A)$ of a subset $A$ of a topological space
is the smallest closed subset containing $A$:
$\bar{A} = \bigcap \{B \mid A \subset B, B \in \mathcal{T}^∗\}$.
**Interior** $\text{Int} A$ of a subset $A$ of a topological space
is the biggest open set contained in $A$:
$\text{Int} A = \bigcup \{S \mid S \subset A, S \in \mathcal{T}\}$.
**Exterior** $\text{Ext} A$ of a subset $A$ of a topological space
is the complement of its closure: $\text{Ext} A = X \setminus \bar{A}$.
**Boundary** $\partial A$ of a subset $A$ of a topological space
is the set of points not in neither its interior nor its exterior,
or equivalently, the set of points in its closure but not in its interior:
$\partial A = X \setminus (\text{Int} A \cup \text{Ext} A) = \bar{A} \setminus \text{Int} A$.
A topological space is the disjoint union of the interior, boundary, and exterior of any subset.
The closure of a subset is the disjoint union of the interior and the boundary of the subset.
For any subset of a topological space, its interior and exterior are open subsets,
and its boundary is a closed subset.

**Limit point** of a subset of a topological space is a point in the space
whose punctured neighbourhoods always intersect with the subset.
A subset is closed if and only if it contains all its limit points (aka derived set $A'$).
The closure of a subset is its union with all its limit points: $\bar{A} = A \cup A'$.
**Isolated point** of a subset of a topological space is a point of the subset
with a punctured neighbourhood disjoint from the set.
Any point of a subset is either a limit point or an isolated point.

A sequence $(x_n)$ of points in a topological space is **eventually in** a subset
if it has a tail contained in the subset: $\exists N: (x_n)_{n \ge N} \subset A$.
A sequence of points in a topological space **converges** to a point in the space
if any neighborhood of the point contains a tail of the sequence:
$\forall U(x) \in \mathcal{T}$, $\exists N \in \mathbb{N}$: $(x_n)_{n \ge N} \subset U$.
We say $(x_n)$ is a **convergent sequence** in the space,
and $x$ is the **limit** of the sequence: $\lim_{n \to \infty} x_n = x$.
Convergent sequences in a Hausdorff space have unique limits.
For a convergent sequence, its limit is its only limit point;
any term not equal to its limit is an isolated point.
Every subsequence of a convergent sequence in a Hausdorff space converges to the same limit.
A sequence of points in a topological space **diverge to infinity**
if every compact subspace contains at most a finite subset of the sequence.
In a first countable Hausdorff space,
a sequence diverges to infinity if and only if it has no convergent subsequence.

Countability properties (first and second countability, separability, the Lindelöf property) ensure that a topological space does not have too many open subsets; separation axioms ($(T_i)_{i=0}^4$; Kolmogorov, Hausdorff, regularity, normality) ensure that it has enough open subsets to conform to our spatial intuition.

**Neighborhood** $U(x)$ of a point $x$ in a topological space
is an open set containing the point: $x \in U(x) \in \mathcal{T}$;
neighborhood of a subset of a topological space is an open set containing the subset.
The term "neighborhood" can be rather misleading:
it does not need to be connected, precompact, or "small" (if relative volume makes sense).
**Neighborhood basis** $B_p$ for a topological space $X$ at a point $p$
is a collection of neighborhoods of $p$
such that every neighborhood of $p$ contains some member of $B_p$.
**First countable** topological space is one where every point has a countable neighborhood basis.
Every metric space is first countable.
**Nested neighborhood basis** $\{U_i\}$ for $X$ at $p$
is a sequence of neighborhoods of $p$ that are successive subsets and form a neighborhood basis.
Every point in a first countable topological space has a nested neighborhood basis.
For a first countable topological space, a subset $A$ is:
open if and only if every sequence converging to a point in $A$ is eventually in $A$;
closed if and only if it contains the limits of every convergent sequence of points in $A$.

**Dense subset** of a topological space is a subset whose closure is the set: $\bar{A} = X$.
**Nowhere dense subset** of a topological space is a subset whose exterior is dense.
**Meager subset** of a topological space is a subset
that can be expressed as a countable union of nowhere dense subsets.
**Separable space** is a topological space containing a countable dense set.
Euclidean spaces are separable.

**Cover** $\mathcal{U}$ of a topological space $X$
is a collection of its subsets whose union is $X$.
**Open cover** is a cover consisting of open subsets.
**Closed cover** is a cover consisting of closed subsets.
**Subcover** $\mathcal{U}'$ of a cover $\mathcal{U}$ of $X$
is a subcollection of $\mathcal{U}$ that still covers $X$.
**Lindelöf space** is a topological space of which every open cover has a countable subcover.

**Second countable** topological space is one with a countable basis for its topology.
A second countable topological space is first countable, separable, and Lindelöf.
For metric spaces, second countability, separability, and the Lindelöf property are equivalent.

**Kolmogorov space** or **T_0-space**
is a topological space where distinct points can be separated by an open set.
**T_1-space** is a topological space where distinct points can be separated by two open sets,
one containing either of the points.
**Hausdorff space** or **T_2-space**
is a topological space where distinct points have disjoint neighborhoods.
The topology of a Hausdorff space is a **Hausdorff topology**.
Metric spaces and almost all topological spaces encountered in analysis are Hausdorff.
**Regular space** or **T_3-space** is a Hausdorff space
where closed subset and exterior point have disjoint neighborhoods.
**Completely regular space** or **Tikhonov space** is a Hausdorff space
where closed subset and exterior point are functionally separable.
**Normal space** or **T_4-space** is a Hausdorff space
where disjoint closed subsets have disjoint neighborhoods.
*Urysohn's Lemma*: Every normal space is completely regular.

**Disconnected** topological space $(X, \mathcal{T})$ is one that can be expressed as
the union of two disjoint, nonempty, open subsets:
$\exists U, V \in \mathcal{T} \setminus \emptyset$: $U \cap V = \emptyset$, $U \cup V = X$.
**Connected** topological space is one that is not disconnected.
A topological space is connected if and only if
no subset is both open and closed except for the empty set and itself:
$\mathcal{T} \cap \mathcal{T}^∗ = \emptyset \cup X$.
**Component** of a topological space $X$ is a maximal nonempty connected subset of $X$.
A topological space is partitioned by its components.
Components of a topological space are closed subsets.
**Path** in a topological space $X$ from a point $p$ to another point $q$
is a continuous map $f: I \mapsto X$ connecting $p$ and $q$: $f(0) = p, f(1) = q$.
**Path-connected** topological space is one where path exists for any pair of points.
If a topological space is path-connected, then it is connected.
**Locally (path-)connected** topological space
is one that admits a basis of (path-)connected open subsets:
every neighborhood of every point contains a (path-)connected neighborhood of the point.
Components of a locally connected topological space are open subsets.
If a topological space is locally path-connected, then it is locally connected.
A locally path-connected space is connected if and only if it is path-connected.

**Compact** topological space is one where every open cover has a finite subcover:
$\mathcal{U} \subset \mathcal{T}$, $\bigcup \mathcal{U} = X$, then
$\exists \{U_i\}_{i=1}^k \subset \mathcal{U}$, $\cup_{i=1}^k U_i = X$.
A subspace of a topological space is compact if and only if
every cover of the subspace by open subsets of the space has a finite subcover.
Note that sometimes a subset of a topological space may be said to have
some property of a topological space such as connectedness or compactness,
which should be understood as the subspace, i.e. the subset endowed with the subspace topology.
Closed subspace of a compact space is compact.
Compact subspace of a Hausdorff space is closed.
Compact subspace of a metric space is bounded.
Compact metric space is complete.
A subspace of a Euclidean space is compact if and only if it is closed and bounded.

**Limit point compact** topological space is one where every infinite subset has a limit point.
Compactness implies limit point compactness.
**Sequentially compact** topological space is one where every sequence has a convergent subsequence.
Sequentially compactness is also known as the Bolzano-Weierstrass property.
For first countable Hausdorff spaces, limit point compactness implies sequential compactness.
For second countable topological spaces, sequential compactness implies compactness.
For metric spaces and second countable Hausdorff spaces,
limit point compactness, sequential compactness, and compactness are equivalent.

**Locally compact** topological space is one
that has a collection of compact subspaces whose interiors cover the space.
**Precompact** or **relatively compact** subspace $A$ of a topological space $X$
is a subspace whose closure $\bar{A}$ is a compact subspace.
For a Hausdorff space, the following are equivalent:
it is locally compact; it has a basis of precompact open subsets;
each point in it has a precompact neighborhood.
A subspace of a locally compact Hausdorff space is again locally compact and Hausdorff,
if it is an open or closed subset.
**Baire space** is a topological space such that
every countable intersection of dense open subsets is dense.
*Baire Category Theorem*:
locally compact Hausdorff spaces and complete metric spaces are Baire spaces.
In a Baire space, the complement of a meager subset is dense.
**Exhaustion** $\{K_n\}$ of a noncompact topological space $X$ by compact subspaces
is a sequence of compact subspaces each includes in its predecessors in its interior
and expands to $X$: $K_n \subset \text{Int}K_{n+1}$, $\bigcup_n K_n = X$.
Locally compact, Hausdorff, second countable space admits exhaustion by compact sets.

**Locally finite** collection of subsets of a topological space
is one such that there is an open cover where each set intersects with
at most a finite subcollection of the collection.
**Refinement** $\mathcal{B}$ of a cover $\mathcal{A}$ of a topological space $X$
is a cover of $X$ such that each set in $\mathcal{B}$ has a superset in $\mathcal{A}$:
$\forall B \in \mathcal{B}, \exists A \in \mathcal{A}: B \subset A$.
**Open refinement** of a cover is a refinement by an open cover.
A topological space $X$ **has finite topological dimension** if there is an integer $k$
such that every open cover has an ("(k+1)-uniformly locally finite") open refinement
such that no point lies in more than $k+1$ of the subsets.
**Topological dimension** of such a topological space is the smallest such integer.
**Paracompact** (仿紧) topological space $X$ is one where
every open cover admits a locally finite open refinement.
Locally compact, Hausdorff, second countable space is paracompact.
Paracompact Hausdorff space is normal.

**Subspace topology** $\mathcal{T}_S$ on a subset $S$ of a topological space $(X, \mathcal{T})$
is the collection of intersections of $S$ with the elements in the topology $\mathcal{T}$:
$\mathcal{T}_S = \{S \cap V \mid V \in \mathcal{T}\}$.
Subspace topology $\mathcal{T}_S$ is a topology on $S$.
Topological **Subspace** $(S, \mathcal{T}_S)$ of a topological space $(X, \mathcal{T})$
is a topological space consisting of a subset $S$ of $X$ and its subspace topology $\mathcal{T}_S$.
Subspace preserves first and second countability, and the Hausdorff property.
A special example of subspace is the subset $\mathbb{Q}$ of rational numbers
in the space of real numbers $\mathbb{R}$ endowed with the Euclidean topology:
it is infinitely disconnected, its components are singletons,
but it does not have the discrete topology and no singleton is open,
and thus it is not locally connected;
it is second countable, Hausdorff, but not locally compact,
because the closure of nonempty open subsets are all noncompact (proof not evident);
it is not a manifold, of course.
**Fiber** $f^{-1}(y)$ over $y$ of a map $f: X \mapsto Y$ between topological spaces
is the subspace that corresponds to the level set of $f$ at $y$.

**Product topology** $\mathcal{T(B_\times)}$ of topologies $\mathcal{T}_X$ and $\mathcal{T}_Y$
is the topology generated by the basis consisting of the product open sets:
$\mathcal{B}_\times = \{U \times V \mid U \in \mathcal{T}_X, V \in \mathcal{T}_Y\}$.
**Product space** $(X \times Y, \mathcal{T(B_\times)})$ of topological spaces $X$ and $Y$
is the topological space consisting of their Cartesian product $X \times Y$
and the product topology $\mathcal{T(B_\times)}$.
Product space preserves first and second countability, the Hausdorff property,
(path-)connectedness, and compactness.

**Disjoint union topology** $\mathcal{T}_\sqcup$
of an indexed family $\{\mathcal{T}_\alpha \mid \alpha \in A\}$ of topologies
is the topology consisting of the disjoint unions of open sets:
$\mathcal{T}_\sqcup = \{\sqcup_\alpha U_\alpha \mid U_\alpha \in \mathcal{T}_\alpha\}$.
**Disjoint union space** $(\sqcup_\alpha X_\alpha, \mathcal{T}_\sqcup)$
of an indexed family $\{(X_\alpha, \mathcal{T}_\alpha) \mid \alpha \in A\}$
of topological spaces is the topological space consisting of
the disjoint union $\sqcup_\alpha X_\alpha$ and the disjoint union topology $\mathcal{T}_\sqcup$.
Disjoint union space preserves first countability, second countability, and the Hausdorff property.

**Quotient topology** (商拓扑) $\mathcal{T}_q$ induced by a surjective map $q: X \mapsto Y$
from a topological space $(X, \mathcal{T})$ onto a set $Y$
is the topology consisting of subsets of $Y$ with open preimages:
$\mathcal{T}_q = \{U \subset Y \mid q^{-1}(U) \in \mathcal{T}\}$.
When the range of a surjective map on a topological space is endowed with the quotient topology,
the map is called a **quotient map**, which is a continuous map.
**Saturated subset** of the domain of a map $f: X \mapsto Y$ between topological spaces
is the preimage of some subset of the codomain: $U = f^{-1}(V), V \subset Y$.
A surjective continuous map is a quotient map if and only if
it takes saturated open/closed subsets to open/closed subsets.
**Quotient space** (商空间) or **identification space** $(q(X), \mathcal{T}_q)$
of a topological space X by an equivalence relation $\sim$---or its canonical projection
q(x) = where is the equivalence class of x w.r.t. $\sim$---
is the topological space consisting of the quotient set $X/\sim$ (set of equivalence classes)
and the quotient topology $\mathcal{T}_q$ induced by the canonical projection.
Quotient space preserves (path-)connectedness and compactness.

**Adjunction space** (黏着空间) $X \cup_f Y$ formed by
attaching a topological space $Y$ to a topological space $X$
along a continuous map $f: A \mapsto X$ on a closed subspace $A$ of $Y$
is the quotient space of the disjoint union of $X$ and $Y$
by equating each point in $A$ with its value:
$X \cup_f Y = q(X \sqcup Y)$, where the canonical projection satisfies
$q(x) = \{x\} \sqcup f^{-1}(x), x \in X$ and $q(y) = \{y\}, y \in Y \setminus A$.
Note that $X \cup_f Y$ is called "the adjunction space formed by attaching $Y$ to $X$ along $f$",
and $f$ is called the **attaching map** (贴映射).
Adjunction space $X \cup_f Y$ is the disjoint union space of the quotient spaces of
$X$ and $Y \setminus A$ under the canonical projection:
$q(X \sqcup Y) = q(X) \sqcup q(Y \setminus A)$.

**Continuous map** between topological spaces at a point
is a map that maps every convergent sequence to the point
to a convergent sequence to its value at the point:
$\lim_{i \to \infty} x_i = x$ then $\lim_{i \to \infty} f(x_i) = f(x)$;
denoted as $\lim_{x_i \to x} f(x_i) = f(x)$.
**Continuous map** $f: X \to Y$ from a topological space $(X, \mathcal{T}_X)$
to a topological space $(Y, \mathcal{T}_Y)$ is a map such that
the preimage of any open set is open: $\forall G \in \mathcal{T}_Y, f^{-1}(G) \in \mathcal{T}_X$.
Equivalently, a continuous map between topological spaces is a map that is continuous at every point.
The set of all continuous maps between two topological spacaes is denoted as $C(X,Y)$ or $C^0(X,Y)$.
**Continuous real-valued function algebra** $C(X)$ on a Hausdorff space
is the algebra consisting of the set of all continuous real-valued functions
endowed with scalar multiplication, pointwise addition, and pointwise multiplication;
i.e. the continuous real-valued function space endowed with pointwise multiplication.
Continuous map preserves (path-)connectedness, compactness, the Baire property.
*Intermediate Value Theorem*:
The range of a continuous real-valued function on a connected topological space is an interval.
*Extreme Value Theorem*:
The range of a continuous real-valued function on a compact topological space
has a maximum and a minimum.

**Bump function** for a subset of a topological space
is a continuous extension of the indicator function on the subset:
$f \in C(X, I)$, $f|_A = 1$.
**Functionally separable subsets** in a topological space
are two subsets such that there is a bump function for one subset that equals zero on the other:
$\exists f \in C(X, I)$: $f|_A = 0$, $f|_B = 1$.
A set of continuous functions on a Hausdorff space **separates points** in the space if
every pair of distinct points can be separated by a function in the set:
$x, y \in X$, $x \ne y$ then $\exists f \in C(X)$: $f(x) \ne f(y)$.
If a set of continuous functions on a Hausdorff space separates points in the space,
then the preimages of open subsets on the real line generates the topology of the space:
$\mathcal{T}\{f^{-1}(U) : f \in A, U \in \mathcal{T}_{\mathbb{R}}\} = \mathcal{T}_X$.
**Support** $\text{supp} f$ of a continuous real-valued function on a topological space
is the closure of the subset where the function is not zero:
$\text{supp} f = \overline{X \setminus f^{-1}(0)}$.
Every closed subset of a normal topological space has a bump function
supported in an arbitrary neighborhood of the subset.

**Exhaustion function** for a topological space $(X, \mathcal{T})$
is a continuous real-valued function whose sublevel sets $f^{-1}(-\infty, c]$ are compact.

**Partition of unity** $\{\psi_\alpha \mid \alpha \in A\}$ on a topological space $X$
subordinate to an indexed open cover $\mathcal{X} = \{X_\alpha \mid \alpha \in A\}$ of $X$
is an indexed family of continuous non-negative functions
suported within the corresponding element of the cover
that add up to one in locally finite sums:
$\sum_\alpha \psi_\alpha (x) = 1$; $\text{supp}\psi_\alpha \subset X_\alpha$;
$\forall p \in X, \exists U(p) \in \mathcal{T}$: $\exists B \subset A, |B| < \infty$,
$\sum_{\alpha \in B} \psi_\alpha (x) = 1$.
For a paracompact Hausdorff space, every open cover admits a subordinate partition of unity.

**Topological isomorphism** (拓扑同构; isomorphisms in the category of topological spaces),
or **homeomorphism** (同胚), is a continuous map between two topological spaces
that has a continuous inverse map.
Two topological spaces are **topologically equivalent**, or **homeomorphic**,
if there is a homeomorphism between them.
**Topological invariant** of a topologically equivalent class of topological spaces
is a property common to all these spaces,
e.g. number of components, (path-)connectedness, compactness, the Baire property, etc.
To prove that two topological spaces are homeomorphic
is usually straightforward, i.e. through definition, by constructing a homeomorphism;
To prove that two topological spaces are not homeomorphic
is usually by showing that they have different topological invariants.

Topological **embedding** (嵌入) $f: X \to Y$ of a topological space into another topological space
is an injective continuous map that yields a homeomorphism
between its domain and its range with the subspace topology.
An embedding is a representation of a topological space in another topological space,
which preserves its connectivity or algebraic properties.
**Reach** $r$ of a topological space embedded in a Euclidean space
is the largest real number such that any point with a distance less than the number from the space
has a unique projection on the space.
Reach can be zero for non-smooth manifolds, such as the graph of $|x|$, aka "corner"
(it is the boundary of the epigraph of $|x|$, where the origin is the corner point).

**Section** (截面) $\sigma$ of a surjective continuous map $\pi: M \mapsto N$
is a continuous map $\sigma: N \mapsto M$ whose composition with $\pi$ is the identity map on $N$:
$\pi \circ \sigma = \text{Id}_N$.
Equivalently, a section of a surjective continuous map is a continuous right inverse of the map.
**Local section** of a continuous map is a section on an open subset of the codomain:
$\sigma: U \mapsto M$, $U \subset N$, $\pi \circ \sigma = \text{Id}_U$.

**Evenly covered** open subset of the codomain of a continuous map
is one whose preimage is a disjoint union of connected open subsets of the domain,
the restriction of the map to each is a homeomorphism to the subset.
Every evenly covered open subset is connected.
**Sheet of the covering** over an evenly covered open set is a component of the preimage of the set.
**Covering map** $q: E \mapsto X$ is a surjective continuous map
on a connected and locally path-connected topological space
such that the codomain has an open cover consisting of evenly covered sets.
For a covering map, the domain is called the **covering space** of the codomain,
and the codomain is called the **base** of the covering.
**Lift** $\tilde\phi: Y \mapsto E$ of a continuous map $\phi: Y \mapsto X$,
given a covering map $q: E \mapsto X$ onto its codomain,
is a continuous map that equals the original map when followed by the covering map:
$q \circ \tilde\phi = \phi$.
The lift of a continuous map from a connected topological space, given a covering map,
is uniquely determined by its value at any one point:
$\exists p \in Y$, $\tilde\phi_1(p) = \tilde\phi_2(p)$ then $\tilde\phi_1 = \tilde\phi_2$.
**Lifting problem** for covering maps is the problem of deciding
whether a continuous map admits a lift to a covering space of its codomain.

**Closed map** between topological spaces is a map that maps closed subsets to closed subsets:
$A \mathcal{T}_X^∗$ then $f(A) \in \mathcal{T}_Y^∗$.
*Closed Map Lemma*:
A continuous map from a compact space to a Hausdorff space is a closed map; in addition,
if it is surjective, it is a quotient map;
if it is injective, it is a topological embedding;
if it is bijective, it is a homeomorphism.

**Proper map** between topological spaces is a map such that
the preimage of a compact subset is compact.
Proper map preserves divergence to infinity.
**Compactly generated** topological space $X$ is one such that
if the intersection of a subset $A$ with any compact subset $K$ is closed in $K$
then $A$ is closed in $X$.
A proper continuous map to a compactly generated Hausdorff space is a closed map.

**Open map** between topological spaces is a map that maps open subsets to open subsets.
For maps between topological spaces, openness, closedness, and continuity are independent.

Manifold builds on a Euclidean space to obtain new spaces that are topologically distinct to the Euclidean space (or its regular domains). Manifolds with the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis.

Topological **manifold** without boundary is a topological space $(M, \mathcal{T})$ that is
Hausdorff, second-countable, and locally Euclidean of a certain dimension $n$, i.e.
has an open cover where each set is homeomorphic to (an open subset/ball of)
the n-dimensional Euclidean space $\mathbb{R}^n$.
The definitions of manifold are equivalent
whether using open subset, open ball, or the Euclidean space itself.
Here $n$ is called the **dimension** $\dim M$ of the manifold.
A topological manifold of dimension $n$ is also known as a topological **n-manifold**.
Open subsets of the Euclidean n-space are n-manifolds.
**n-manifold topology** is the topology of an n-manifold.

**Closed n-dimensional upper half-space** $\mathbb{H}^n$
is the subspace of the Euclidean n-space with non-negative n-th component:
$\mathbb{H}^n = \{x \in \mathbb{R}^n \mid x_n \ge 0\}$.
**n-manifold with boundary** is a topological space that is Hausdorff, second countable,
and has an open cover where some sets are homeomorphic to (an open subset of) the Euclidean n-space
and the others are homeomorphic to (an open subset of) $\mathbb{H}^n$.
Most propositions true for boundaryless manifolds are also true for manifolds with boundary,
perhaps with minor modifications;
unless explicitly stated, manifolds in this article refers to those with and without bounary.

Dimension is a topological invariant of a manifold. An n-manifold has topological dimension $n$.

Manifold has many nice properties for a topological space: manifold is locally path-connected, locally compact, paracompact, and normal; any subset of a manifold is compactly generated; the fundamental group of a manifold is countable; (smooth) n-manifold admits a (smooth) embedding into the Euclidean 2n-space. Compact manifolds can be embedded into closed and bounded subsets of some Euclidean space.

Any closed subset $A$ of a manifold $M$ can be the zero set of a continuous non-negative function $f$ on the manifold: $f^{-1}(0) = A$. Strengthening Urysohn’s Lemma, any closed subset $B$ disjoint to $A$ can be made the preimage of one, while other points take values in between: $f^{-1}(1) = B$, $f(M) = I$. Manifold admits a positive exhaustion function.

The empty set is a manifold of any dimension. Countable discrete spaces $(N, 2^N)$, $N \subset \mathbb{N}$, are all the 0-manifolds. A rubber band is a 1-manifold; a broken rubber band is a 1-manifold with 0-dimensional boundary. The unit circle $\mathbb{S}^1$ is a 1-manifold; the unit interval $I = [0, 1]$ is a 1-manifold with 0-dimensional boundary. A balance ball is a 2-manifold; half a balance ball is a 2-manifold with 1-dimensional boundary. The unit sphere $\mathbb{S}^2$ is a 2-manifold; the closed unit disk $\bar{\mathbb{B}}^2$ is a 2-manifold with 1-dimensional boundary. Cylinder $\mathbb{S}^1 \times I$ and the Möbius band are 2-manifolds with 1-dimensional boundaries. The torus $\mathbb{T}^2$ and the Klein bottle $K$ are 2-manifolds. The projective plane $\mathbb{P}^2$ is a 2-manifold. A medicine ball is a 3-manifold with 2-dimensional boundary.

Real **projective space** $\mathbb{P}^n$ (or $\mathbb{RP}^n$) of dimension $n$
is the quotient space of $\mathbb{R}^{n+1} \setminus \{0\}$
by the projection map $q(x) = \{ax \mid a \ne 0\}$ equating points up to nonzero scaling.
Note that $\mathbb{P}^n$ is not homeomorphic with $\mathbb{S}^n$
because opposite/antipodal points are equivalent.
The n-dimensional real projective space $\mathbb{P}^n$ is an n-manifold,
and it is homeomorphic to the n-sphere via the covering map
$\text{span}: \mathbb{S}^n \mapsto \mathbb{P}^n$
(for n=1, trivial proof; for n>1, [@Lee2011, Cor 11.33]).
**Complex projective space** $\mathbb{CP}^n$ is the quotient space of
$\mathbb{C}^{n+1} \setminus \{0\}$ by equating points up to nonzero scaling.
The n-dimensional complex projective space $\mathbb{CP}^n$ is a 2n-manifold.

The state space of a dynamical system is often considered a manifold (literally, the set of all possible values of a variable with certain constraints), which can be much more complex than a Euclidean space due to conservation laws or other constraints. The dimension of the manifold corresponds to the degrees of freedom of the system, where the points are specified by generalized coordinates. (The configuration space of double pendulum is a 2-torus.) Applications: symplectic manifold for analytical mechanics (Lagrangian, Hamiltonian) [@Arnold1989]; Lorentzian 4-manifold for general relativity; complex manifold for complex analysis.

Local **coordinate chart** $\phi: U \to \mathbb{R}^n$ of an n-manifold
or **interior chart** of an n-manifold with boundary
is a homeomorphism from an open subset of the manifold to an open subset of $\mathbb{R}^n$.
**Boundary chart** $\phi: U \to \mathbb{H}^n$ of an n-manifold with boundary
is a homeomorphism from an open subset of the manifold to an open subset of $\mathbb{H}^n$
that includes a part of the boundary of the half-space.
**Interior point** of a manifold is a point in the domain of some interior chart.
**Boundary point** of a manifold with boundary
is a point in the domain of a boundary chart that sends it to the boundary of the half-space.
**Interior** $\text{Int} M$ of a manifold is the set of all its interior points.
The interior of an n-manifold is an open subset of the manifold
and an n-manifold without boundary.
**Boundary** $\partial M$ of a manifold with boundary is the set of all its boundary points.
The boundary of an n-manifold with boundary is a closed subset of the manifold
and an (n-1)-manifold without boundary.
Any manifold is the disjoint union of its interior and boundary.
The boundary of a manifold as a topological space and as a subset of a topological space
are independent concepts.

**Local coordinates** $(x_i)_{i=1}^n$ on the domain of a local coordinate chart
are the component functions of the chart: $\phi = (x_i)_{i=1}^n$.
**Local coordinate representation** $\hat{p}$ of a point $p$ in the domain of a chart
is the tuple of local coordinate values at the point: $\hat{p} = \phi(p)$.
**Atlas** $\mathcal{A}$ of a manifold $M$ is a class of coordinate charts whose domains cover $M$.
**Transition map** from a coordinate chart $\phi_1$ to another coordinate chart $\phi_2$
of the same n-manifold, whose domains $U_1$ and $U_2$ overlap,
is the composite map $\phi_2 \circ \phi_1^{-1}: \phi_1(U_1 \cap U_2) \to \mathbb{R}^n$.
Note that transition maps are functions between subsets of $\mathbb{R}^n$.

**Coordinate domain** $U$ is the domain of a local coordinate chart.
A manifold admits a finite atlas; the coordinate domains are disconnected in general.
**Coordinate ball** $B$ of an n-manifold is an open subset of the manifold
that is homeomorphic to an open ball $B_r(0)$ in $\mathbb{R}^n$.
**Coordinate half-ball** of an n-manifold with boundary is an open subset of the manifold
that is homeomorphic to an open half-ball $B_r(0) \cap \mathbb{H}^n$ in $\mathbb{H}^n$.
**Regular coordinate ball** of an n-manifold
is one such that there is a chart on a larger coordinate ball
that maps it to an open ball and maps its closure to the closed ball:
$\phi: B' \mapsto B_{r'}(x)$, $\phi(B) = B_r(x)$, $\phi(\bar{B}) = \bar{B}_r(x)$.
A manifold has a countable basis of regular coordinate balls.
The complement $M \setminus B$ of a regular coordinate ball $B$
in an n-manifold $M$ without boundary is an n-manifold with boundary
and the boundary is homeomorphic to the (n-1)-sphere $\mathbb{S}^{n-1}$.

**Global coordinate chart** is a coordinate chart on the entire manifold.
Each point of a manifold is in the domain of some local coordinate chart,
but the manifold may not have a global coordinate chart.
In fact, an n-manifold has a global coordinate chart if and only if
it is homeomorphic to an open subset or a regular domain of $\mathbb{R}^n$.
For example, angular coordinate is a chart of a circle, but not a global homeomorphism;
in the same sense, geographical coordinates is a local chart of a sphere.

Table: Abstract Object vs Representations

Abstract Object | Representation | Reference | Invariants |
---|---|---|---|

vector space |
Cartesian power of field | basis | dimemsion, field |

vector | tuple | basis | |

linear operator | matrix | two bases | singular value-space pairs |

tensor | array | n bases | |

linear transformation | square matrix | basis | eigenvalue-eigenspace pairs |

inner product space |
Euclidean space | basis | length, angle |

orthogonal linear trans | normal matrix | - | orthogonal eigen-basis |

linear isometry | unitary matrix | - | length, angle |

self-adjoint | Hermitian matrix | - | real eigenspaces |

scaling | positive definite matrix | - | eigen-direction |

strongly connected graph | irreducible nonnegative m | - | max eigenvalue-eigenvector |

manifold |
coordinate balls | atlas | topological invariants |

point | coordinate | chart | |

continuous map | continuous function | chart | |

smooth manifold |
embedded submanifold | embedding | smooth invariants |

tangent vector | tuple | chart | |

differential | Jacobian matrix | charts | |

Riemannian manifold |
geometric invariants | ||

isometry | unitary Jacobian matrix | charts | distance, volume, curvature |

Manifolds are abstract objects that do not come with any predetermined choice of coordinates, and unlike subsets of a Euclidean space, their points often do not have canonical representation. Reversely, a manifold can be identified as a subset of a higher-dimensional Euclidean space, and then an atlas of charts covering this subset is constructed. Manifolds can be also constructed by surgery, i.e. specifying an atlas which is itself defined by transition maps.

Objects defined globally on a manifold are also abstract,
and thus shall not depend on a particular choice of coordinates.
**Invariant definition** of an abstract object is a definition that is representation-independent.
An abstract object on a manifold can be defined invariantly, if possible;
one can also give it a coordinate-dependent definition
and then prove that the definition gives the same results in any coordinate chart.
For example, a manifold may be endowed with more structures besides a nice topology,
such as smooth structure (see Smooth Manifold / Differential Topology)
and geometric structure (see Differential Geometry).
Such structures are first defined on each chart separately,
and if all the transition maps are compatible with the structure,
it transfers to the manifold.

Subspace of a manifold may not be a manifold;
for example, cross lines $y^2 - x^2 = 0$, opposite cones $y^2 - x^2 \ge 0$.
Recall that the graph $\Gamma(f)$ of a map $f: X \mapsto Y$ is the subset of $X \times Y$
defined as $\Gamma(f) = \{(x, f(x)) \mid x \in X\}$.
The graph of any continuous function from any open subset of $\mathbb{R}^n$,
endowed with the subspace topology, is an n-manifold homeomorphic to the domain:
$\forall U \in \mathcal{T}(\mathbb{R}^n)$, $\forall f \in C^0(U, \mathbb{R}^k)$,
$\Phi_f: U \mapsto \Gamma(f)$, $\Phi_f(x) = (x, f(x))$ is a homeomorphism.
Projection onto the first $n$ components is a global coordinate chart of the manifold,
known as the **graph coordinates**: $\phi = (\pi_i)_{i=1}^{n}|_{\Gamma(f)}$.

The product space $M \times N$ of an m-manifold and an n-manifold, both without boundary,
is an (m+n)-manifold without boundary, which may be call a **product manifold**.
**n-torus** is the n-th power of 1-spheres: $\mathbb{T}^n = (\mathbb{S}^1)^n$;
which is often considered as the subset of the Euclidean 2n-space defined by
$\forall i \in \{i\}_{i=1}^n$, $(x^{2i-1})^2 + (x^{2i})^2 = 1$.
A finite cylinder $\mathbb{S}^1 \times I$.

Disjoint union of n-manifolds is an n-manifold. Disjoint union of manifolds of different dimensions are not manifolds.

Quotient space of a manifold may be a manifold.
For example, closed unit disk $\bar{\mathbb{B}}^2$ to sphere $\mathbb{S}^2$.
**Polygonal region** is a compact subset of $\mathbb{R}^2$ whose boundary is a polygon.
The quotient space of a disjoint union of polygonal regions
that identifies pairs of edges in affine homeomorphisms
(affine maps that identify the two pairs of vertices in either order)
is a compact 2-manifold.
For example, square $I^2$ to cylinder $\mathbb{S}^1 \times I$ by $(0, t) \sim (1, t)$;
square to Möbius band by $(0, t) \sim (1, 1-t)$;
cylinder $\mathbb{S}^1 \times I$ to torus $\mathbb{T}^2$ by $(t, 0) \sim (t, 1)$;
cylinder to Klein bottle $K$ by $(t, 0) \sim (1-t, 1)$.

If $M$ and $N$ are n-manifolds with boundary
and $h: \partial N \mapsto \partial M$ is a homeomorphism between their boundaries,
the adjunction space $M \cup_h N$ (formed by attaching $N$ to $M$ along $h$) is called
"an adjunction space formed by attaching $M$ and $N$ together along their boundaries",
which is an n-manifold without boundary, referred to as an "**adjunction manifold**" hereafter.
The original manifolds can be embedded into an adjunction manifold as closed subsets
which cover the adjunction manifold and intersect at the attached boundary.
**Double** $D(M)$ of a manifold $M$ with boundary is the adjunction manifold formed by
attaching two copies of $M$ together along the identity map of its boundary:
$D(M) = M \cup_h M$, $h = \text{Id}_{\partial M}$.
**Connected sum** $M_1 \# M_2$ of connected n-manifolds $M_i$, $i = 1, 2$,
is a connected adjunction n-manifold formed by cutting out a regular coordinate ball $B_i$ on each
and attaching the remainders $M'_i = M_i \setminus B_i$ together along their boundaries:
$M_1 \# M_2 = M'_1 \cup_f M'_2$, $f: \partial M'_2 \mapsto \partial M'_1$ is a homeomorphism.
The Klein bottle is homeomorphic to the connected sum of two projected planes.
The connected sums of two given n-manifolds
are in at most two homeomorphic classes which differ only in orientation.

**Orientable** compact surface is one that admits an oriented presentation.
The Möbius band is not orientable.
A compact connected surface is orientable if and only if
it is homeomorphic to the sphere or a connected sum of one or more tori,
or equivalently, it does not contain a subset homeomorphic to the Möbius band.
Orientability of a compact surface is a topological invariant.
Orientable **surface of genus 0** is the sphere.
**Orientable surface of genus n** is the connected sum of $n$ tori.
**Nonorientable surface of genus n** is the connected sum of $n$ projective planes.
The genus (亏格) of a surface can be recovered from the fundamental group,
so it is a topological invariant.

Orientable triangulated manifolds can also be defined...

Classification of 1-manifolds. Every connected 1-manifold is homeomorphic to exactly one of the following: (1) the circle $\mathbb{S}^1$: compact, no boundary; (2) the real line $\mathbb{R}$: noncompact, no boundary; (3) the unit interval $[0, 1]$: compact, with boundary; (4) closed half line $[0, \infty)$: noncompact, with boundary. Although compact 1-manifolds are homeomorphic to the circle, their embeddings may not be isotopic, e.g. nontrivial knots vs the unknot.

Classification of 2-manifolds (aka **surfaces**).
Every compact, connected 2-manifold without boundary
is homeomorphic to exactly one of the following:
(1) The sphere $\mathbb{S}^2$;
(2) A connected sum of one or more copies of the torus $\mathbb{T}^2$;
(3) A connected sum of one or more copies of the projective plane $\mathbb{P}^2$.

Poincaré conjecture [@Poincare1904]: Simply connected, compact 3-manifolds are homeomorphic to the 3-sphere $\mathbb{S}^3$. Stephen Smale proved analogous cases for $n≥5$ [@Smale1961]; Michael Freedman proved the analogous case for $n=4$ [@Freedman1982]; Grigori Perelman completed Richard Hamilton's program which uses Ricci flow to prove the existence of geometric decompositions, and thus proved the Thurston geometrization conjecture, which implies the Poincaré conjecture [@Perelman2006].

Thurston geometrization conjecture [@Thurston1982]: Every compact, orientable 3-manifold can be expressed as a connected sum of compact manifolds, each of which either admits a Riemannian covering by a homogeneous Riemannian manifold or can be cut along embedded tori so that each piece admits a finite-volume locally homogeneous Riemannian metric.

No program can decide whether two smooth n-manifolds, $n≥4$, are diffeomorphic. [@Markov1958] It is generally undecidable whether two n-manifolds, $n>4$, are homeomorphic.

**Differential topology** is the study of
the invariant properties of smooth manifolds under smooth deformations (diffeomorphisms).
Smoothness is the notion for a topological space to have (co-)tangent spaces that vary continuously,
and sucessively higher order differentials that also vary continuously.
The smoothness condition, i.e. infinite continuous differentiability,
can often be relaxed in practice, because continuous maps
(and finitely continuously differentiable maps) can be uniformly approximated by smooth maps.
Smooth manifolds admits differentiation and integration, see Calculus on Manifolds.

**Smoothly compatible** coordinate charts of a manifold are two charts $\phi_1, \phi_2$
such that either their domains overlap and their transition maps $\phi_j \circ \phi_i^{-1}$
are smooth functions, or their domains are disjoint.
**Smooth atlas** of a manifold is an atlas where any two charts are smoothly compatible.
**Smooth structure**, **C^∞ structure**, or **maximal smooth atlas** of a manifold
is a smooth atlas that is not properly contained in any larger smooth atlas,
i.e. any chart that is smoothly compatible with every chart in the atlas is already in it.
**Smooth chart** is any chart in a smooth structure.
**Smooth structure determined by a smooth atlas** of a manifold
is the unique maximal smooth atlas $\bar{\mathcal{A}}$ containing that smooth atlas.
If a manifold can be covered by a single chart $\phi$,
then $\phi$ forms a singleton atlas $\mathcal{A} = \{\phi\}$,
which is a smooth atlas and thus determines a smooth structure on the manifold.
Two smooth atlases for a manifold determine the same smooth structure
if and only if their union is a smooth atlas.

**Smooth manifold** $(M, \mathcal{T}, \mathcal{A})$ (sometimes called "differentiable manifold")
is a manifold $M$ endowed with a smooth structure $\mathcal{A}$.
The graph of any smooth function from any open subset of $\mathbb{R}^n$,
endowed with the subspace topology and the smooth structure determined by
the smooth atlas consisting of the graph coordinate chart,
is a smooth n-manifold diffeomorphic to the domain.
(See Regular Level Set Theorem for a generalization.)
**Smooth manifold structure** $(\mathcal{T}, \mathcal{A})$
is a manifold topology together with a smooth structure.

Analogous to smooth structure, **C^k structure**,
**real-analytic structure** or **C^ω structure**, and **complex-analytic structure** are defined
such that the transition maps between overlapping coordinate charts are, respectively,
$C^k$ (weaker than smooth) and real- and complex-analytic (stronger than smooth).
Manifold endowed with such a structure is called
**C^k manifold**, **real-analytic manifold**, and **complex manifold**, respectively.
Note that $C^0$ manifold is equivalent to topological manifold.

**Smooth structure** of a manifold with boundary is a smooth structure containing boundary charts.
**Smooth manifold with boundary** is a manifold with boundary endowed with a smooth structure.
**Smooth chart** for a manifold with boundary is any chart in its smooth structure.
**Regular coordinate half-ball** of an n-manifold with boundary
is one such that there is a smooth chart on a larger coordinate half-ball
that maps it to an open half-ball and maps its closure to the closed half-ball:
$\phi: B' \mapsto B_{r'}(0) \cap \mathbb{H}^n$, $\phi(B) = B_r(0) \cap \mathbb{H}^n$,
$\phi(\bar{B}) = \bar{B}_r(0) \cap \mathbb{H}^n$.

For the nonnegative subspace $\bar{\mathbb{R}}_+^n$ of the Euclidean n-space,
its **boundary point** is a point where at least one coordinate vanish;
its **corner point**, if $n \ge 2$, is a point where at least two coordinates vanish.
The number of vanished coordinates can be considered as the "type" of a boundary (or corner) point.
**Chart with corners** $\phi: U \to \bar{\mathbb{R}}_+^n$ for an n-manifold with boundary
is a coordinate chart to an open subset of $\bar{\mathbb{R}}_+^n$.
A chart with corners may contain any type of boundary points in its range, or none at all.
Charts with corners containing different types of boundary points cannot be smoothly compatible.
**Smooth structure with corners** of a manifold with boundary
is a smooth structure consisting of charts with corners.
**Smooth manifold with corners**
is a manifold with boundary endowed with a smooth structure with corners.
**Smooth chart with corners** is any chart with corners in a smooth structure with corners.
*Invariance of Corner Points*:
If a point in a smooth manifold with corners
is mapped to a corner point of the nonnegative subspace for some smooth chart with corners, then
it is mapped to a corner point (of the same type) in all smooth charts with corners containing it.
**Boundary point** of a smooth manifold with corners
is the preimage of a boundary point of the nonnegative subspace in a smooth chart with corners.
**Corner point** of a smooth manifold with corners
is the preimage of a corner point of the nonnegative subspace in a smooth chart with corners.
A smooth manifold with corners is a smooth manifold with or without boundary
if and only if it has no corner points.
The boundary of a smooth manifold with corners is in general not a smooth manifold with corners;
e.g. $\bar{\mathbb{R}}_+^n$ as a submanifold of $\mathbb{R}^n$ is a smooth n-manifold with corners,
but its boundary is merely a finite union of smooth (n-1)-manifolds with corners.
Most constructs for smooth manifolds with or without boundary
can also be defined on smooth manifolds with corners in the same way,
using smooth charts with corners in place of smooth boundary charts.

**Subspace smooth structure** $\mathcal{A}_S$
on a subspace $S$ of a smooth manifold $(M, \mathcal{A})$
is the collection of smooth charts restricted to $S$:
$\mathcal{A}_S = \{\phi|_S: \phi \in \mathcal{A}\}$.

**Product smooth structure** is the smooth structure determined by the smooth atlas
consisting of products $\phi_1 \times \phi_2$ of smooth charts of component manifolds.
The product manifold $M \times N$ of a smooth manifold without boundary and
a smooth manifold with boundary is a smooth manifold with boundary,
and the boundary is the Cartesian product of the former and the boundary of the latter:
$\partial (M \times N) = M \times \partial N$.

**Collar neighborhood** of the boundary of a smooth manifold $M$ is a neighborhood of the boundary
and is the image of a smooth embedding $F: [0,1) \times \partial M \mapsto M$
that restricts to the identification on the boundary: $F(0, p) = p$.
Collar neighborhood exists.
If $M$ and $N$ are smooth n-manifolds with boundary
and $h: \partial N \mapsto \partial M$ is a diffeomorphism between their boundaries,
the adjunction manifold $M \cup_h N$ endowed with a smooth structure such that
the original manifolds are each diffeomorphic to a regular domain in the adjunction manifold
which cover the adjunction manifold and intersect at their boundaries,
is a smooth n-manifold without boundary, referred to as a "**smooth adjunction manifold**".
**Smooth connected sum** of connected smooth n-manifolds
is a connected smooth adjunction n-manifold
formed just like connected sum but by a diffeomorphism.

*Smooth Manifold Chart Lemma*:
A set $X$ can be endowed with an n-manifold topology and a smooth structure,
defined by $\mathcal{T}_\phi = \mathcal{T}(\{ \phi_\alpha^{-1}(V) :
V \in \mathcal{T}_{\mathbb{R}^n}, \alpha \in A\})$
and $\mathcal{A}_\phi = \overline{\{\phi_\alpha\}_\alpha}$,
given a collection of maps $\phi_\alpha: U_\alpha \mapsto \mathbb{R}^n$ that satisfy the following:
(1, 2) homeomorphism: each map is an injection to an open set,
and maps the intersection of its domain and another to an open set;
(3) smooth compatibility: each composite map $\phi_\beta \circ \phi_\alpha^{-1}$ is smooth;
(4) second countablity: the domains of a countable subcollection cover $X$;
(5) Hausdorff property: distinct points not in one domain are in disjoint domains.
**Grassmann manifold** $G_k(V)$ is the set of all k-dimensional linear subspaces
of an n-dimensional real vector space $V$, endowed with certain smooth structure,
is a smooth k(n-k)-manifold.
If $V$ is the Euclidean space $\mathbb{R}^n$, Grassmann manifold is denoted as $G_{k,n}$.
Projective spaces are special cases of Grassmann manifolds: $G_{1,n+1} = \mathbb{P}^n$.

**Local coordinate representation** $\hat{F}: \phi_1(U_1) \mapsto \phi_2(U_2)$
of a map $F: M_1 \mapsto M_2$ between smooth manifolds
w.r.t. smooth charts $\phi_1$ and $\phi_2$ on domains $U_1$ and $U_2$, $F(U_1) \subset U_2$,
is the map defined by $\hat{F} = \phi_2 \circ F \circ \phi_1^{-1}$.
**Smooth map** $F: M_1 \mapsto M_2$ between smooth manifolds
is a map with smooth local coordinate representations everywhere.
The set of all smooth maps from $M$ to $N$ is denoted $C^\infty(M, N)$.
The set of all smooth real-valued functions on a manifold $M$ is denoted $C^\infty(M)$,
which is a vector space over field $\mathbb{R}$.

A smooth map can be constructed from smooth maps on subsets of the domain
by either gluing or blending them together.
*Gluing Lemma for Smooth Maps*:
Given smooth maps $F_\alpha: U_\alpha \mapsto N$ on open submanifolds of a smooth manifold $M$
that cover the manifold and the maps are identical on overlapping domains,
then there is a unique smooth map $F: M \mapsto N$ matching the maps on their domains.
**Smooth partition of unity** on a smooth manifold
is a partition of unity consisting of smooth functions.
Smooth partition of unity exists for any smooth manifold $M$ and any indexed open cover of $M$.
Any smooth function $f: A \mapsto \mathbb{R}^k$ on a closed subset $A$ of a smooth manifold $M$
can be extended to a smooth function $\tilde{f}$ on $M$
that vanishes on any open subset $U$ containing $A$.
Smooth manifold admits a smooth positive exhaustion function.
Closed subset $A$ of a smooth manifold $M$ can be the zero set of
a smooth non-negative function $f$ on the manifold: $f^{-1}(0) = A$.

**Smooth isomorphism** or **diffeomorphism** (微分同胚) between smooth manifolds
is a smooth map with a smooth inverse.
Two smooth manifolds are **smoothly equivalent** or **diffeomorphic**, denoted as $\approx$,
if there is a diffeomorphism between them.
Note that although the real projective n-space and the n-sphere are homeomorphic,
they are diffeomorphic if and only if n = 1.
The complex projective 1-space and the 2-sphere are diffeomorphic.
The complex projective n-space and $\mathbb{S}^{2n+1} / \mathbb{S}^1$ are diffeomorphic.

**Smooth invariant** or **diffeomorphic invariant** of a smoothly equivalent class
of smooth manifolds is a property common to all these manifolds.
Diffeomorphism is a subclass of homeomorphism,
and smooth invariants are not necessarily topological invariants.

**Derivation** (导数) $v: C^\infty(M) \mapsto \mathbb{R}$ at a point $p$ of a smooth manifold $M$
is a functional on smooth real-valued functions
that is linear over $\mathbb{R}$ and satisfies the product rule at $p$:
$\forall a, b \in \mathbb{R}$, $\forall f, g \in C^\infty(M)$,
$v (a f + b g) = a (v f) + b (v g)$, $v (f g) = (v f) g + f (v g)$.
**Tangent space** $T_p M$ at a point $p$ on a smooth manifold $M$
is the set of all derivations at $p$.
Tangent space $T_p M$ is a vector space of dimension $n = \dim M$,
and derivations at $p$ are thus also called **tangent vectors** at $p$.
The canonical isomorphism $F: V \mapsto T_a V$ from a vector space to its tangent space at a point
is writen as $F(v) = D_v|_a$, defined by $D_v|_a f = \frac{d}{dt}\Bigg{|}_{t=0} f(a+tv)$.
In particular, $\mathbb{R}^n \cong T_a \mathbb{R}^n$ by such identification.
The tangent space to a product manifold can be identified with
the direct sum of the tangent spaces of the component manifolds:
$T_{(p,q)} (M \times N) \cong T_p M \oplus T_q N$.

**Coordinate vector** (坐标切向量) $\partial / \partial x^i |_p$
at a point $p$ on a smooth manifold $M$ associated with a smooth chart containing $p$
is the tangent vector at $p$ that equals the i-th partial derivative of
the coordinate representation of any function at the coordinate representation of the point:
$\forall f \in C^\infty(M)$,
$\frac{\partial}{\partial x^i} \bigg{|}_p f = \frac{\partial \hat f}{\partial x^i} (\hat p)$.
**Coordinate basis** $(\partial / \partial x^i |_p)_{i=1}^n$ for a tangent space $T_p M$
associated with a smooth chart containing $p$
is the n-tuple of the coordinate vectors at $p$ associated with the chart.
A coordinate basis for a tangent space is an algebraic basis.
**Component function** $v^i$ of a tangent vector $v$ w.r.t. a coordinate basis
is the value of $v$ applied to the i-th coordinate function: $v^i = v(x^i)$.
**Local coordinate representation** $\hat{v}$ of a tangent vector $v$ w.r.t. a coordinate basis
is the n-tuple of its components: $\hat{v} = (v^i)_{i=1}^n$.
Tangent vector can be written uniquely as a linear combination of
a coordinate basis and its local coordinate representation wherein:
$v = v^i \frac{\partial}{\partial x^i} \bigg{|}_p$.

Pointwise **differential** $d F_p$ or **tangent map** $T_p F$ of
---or **pushforward** (前推) $F_∗$ of tangent vectors by---a smooth map
$F \in C^\infty(M, N)$ at a point $p \in M$
is a linear operator $d F_p \in \mathcal{L}(T_p M, T_{F(p)} N)$
that takes a tangent vector $v$ at the point to the tangent vector at its value
that equals $v$ acting on the composition of smooth real-valued functions with the map:
$\forall f \in C^\infty(N)$, $d F_p(v)(f) = v(f \circ F)$.
**Local coordinate representation** $\widehat{d F_p}$ of a differential $d F_p$
w.r.t. the coordinate bases for $T_p M$ and $T_{F(p)} N$
associated with smooth charts $\phi$ and $\psi$
is the Jacobian matrix of the coordinate representation of $F$ w.r.t. the charts:
$\widehat{d F_p} = \left[\frac{\partial \hat{F}^j}{\partial x^i} (\hat p)\right]$.
(Note that the columns and rows are the opposite of the convention in Analysis.)
The pushforward of coordinate vectors by a smooth map can be written as:
$d F_p \left(\frac{\partial}{\partial x^i} \bigg{|}_p \right) =
\frac{\partial \hat{F}^j}{\partial x^i} (\hat p) \frac{\partial}{\partial y^j} \bigg{|}_{F(p)}$.

**Tangent bundle** (切丛) $T M$ of a smooth manifold $M$
is the disjoint union of the tangent spaces at all points of $M$:
$T M = \sqcup_{p \in M} T_p M$.
Elements of a tangent bundle are often written as $(p, v)$, point before tangent vector.
Natural **projection map** $\pi: T M \mapsto M$ of a tangent bundle
sends each indexed tangent vector to the point of tangent: $\pi(p, v) = p$.
**Natural coordinates** $(\hat{p}, \hat{v})$
of an element $(p, v)$ of a tangent bundle associated with a smooth chart
is the tuple of the local coordinate representation of $p$ w.r.t. the chart and
that of $v$ w.r.t. the coordinate basis.
By the Smooth Manifold Chart Lemma, the tangent bundle of a smooth n-manifold admits
a natural topology $\mathcal{T}_{\tilde{\phi}}$ and a natural smooth structure
$\mathcal{A}_{\tilde{\phi}}$ determined by the collection of natural coordinate maps
$\tilde{\phi}_\alpha: \pi^{-1}(U_\alpha) \mapsto \mathbb{R}^{2n}$
associated with each smooth chart $\phi_\alpha: U_\alpha \mapsto \mathbb{R}^n$,
so that $(T M, \mathcal{T}_{\tilde{\phi}}, \mathcal{A}_{\tilde{\phi}})$ is a smooth 2n-manifold
and its projection map is a smooth map.
The tangent bundle of a smooth n-manifold with a global smooth chart
is diffeomorphic to the product manifold $M \times \mathbb{R}^n$, a subset of $\mathbb{R}^{2n}$.

**Global differential** $d F$ or **global tangent map** $T F$ of
---or **global pushforward** $F_∗$ of tangent vectors by---a smooth map
is the disjoint union of its pointwise differentials: $d F(p, v) = (F(p), d F_p(v))$.
The global differential of a smooth map is a smooth map itself:
$F \in C^\infty(M, N)$, then $d F \in C^\infty(T M, T N)$.

**Cotangent space** $T_p^∗ M$ at a point on a smooth manifold
is the dual space to the tangent space at that point: $T_p^∗ M = (T_p M)^∗$.
Tangent **covector** at a point is an element of the cotangent space at the point.

**Cotangent bundle** (余切丛) $T^∗ M$ of a smooth manifold
is the disjoint union of its cotangent spaces: $T^∗ M = \sqcup_{p \in M} T_p^∗ M$.
The natural projection map $\pi: T^∗ M \mapsto M$ of a cotangent bundle
sends each indexed tangent covector to the point of tangent: $\pi(p, \omega) = p$.
**Coordinate covector field** $\lambda^i$ or $d x^i$ associated with a smooth chart
are the maps that take each point in the domain of the chart to
the covectors dual to the i-th coordinate vector associated with the chart:
$\langle \lambda^i, \frac{\partial}{\partial x^j} \rangle = \delta^i_j$.
**Cotangent bundle** $(T^∗ M, \pi, \{\Phi_\alpha\}_{\alpha \in A}))$ of a smooth manifold
is the smooth vector bundle of rank $n$ over $M$, following the vector bundle chart lemma,
such that the total space $T^∗ M$ is endowed with the unique topology and smooth structure
for which all coordinate covector fields are smooth local sections,
and the maps sending each indexed covector to its coordinate representation
in the coordinate dual basis are diffeomorphisms:
$\Phi_\alpha (\xi_i \lambda^i|_p) = (p, (\xi_i)_{i=1}^n)$.
**Natural coordinates** $(x^i, \xi_i)$ of an element $(p, \omega)$ of a cotangent bundle
associated with a smooth chart is the tuple of the local coordinate representation of
$p$ w.r.t. the chart and that of $\omega$ w.r.t. the coordinate dual basis.
Natural coordinates are smooth charts of a cotangent bundle.

Pointwise **cotangent map** $d F_p^∗$ of
---or **pullback** (拉回) $F_p^∗$ of tangent covectors by---
a smooth map $F \in C^\infty(M, N)$ at a point $p \in M$
is the dual operator $(d F_p)^∗ \in \mathcal{L}(T_{F(p)}^∗ N, T_p^∗ M)$
of its pointwise differential at that point:
$\forall \omega \in T_{F(p)}^∗ N$, $\forall v \in T_p M$, $d F_p^∗ (\omega)(v) = \omega(d F_p (v))$.
**Global cotangent map** $d F^∗$ of
---or **global pullback** of tangent covectors $F^∗$ by---a diffeomorphism
is the map whose restriction to each cotangent space $T_q^∗ N$ equals $d F_{F^{-1}(q)}^∗$.
The global pullback of a diffeomorphism is a smooth bundle homomorphism
from the cotangent bundle $T^∗ N$ of its codomain to the cotangent bundle $T^∗ M$ of its domain.

**Rank** of a smooth map $F$ at a point $p$, or rank of the differential $d F_p$,
is the rank of the Jacobian matrix of $F$ in any smooth chart containing $p$:
$\text{rank}~d F_p = \dim \text{im}(d F_p)$.
A smooth map has **constant rank** if its rank at every point is the same,
which is denoted as $\text{rank}~F$.

A smooth map has **full rank** if its rank at every point equals the lower of
the dimensions of its domain and its codomain: $\text{rank}~F = \min\{\dim M, \dim N\}$.
**Smooth submersion** (浸没) is a smooth map whose differentials are surjective everywhere:
$\text{rank}~F = \dim N$.
**Smooth immersion** (浸入) is a smooth map whose differentials are injective everywhere:
$\text{rank}~F = \dim M$.
**Local diffeomorphism** $F: M \mapsto N$ is a smooth map
that equals gluing together some diffeomorphisms whose ranges are open subsets.
A map between smooth manifolds without boundary is a local diffeomorphism
if and only if it is both a smooth submersion and a smooth immersion.
*Rank Theorem* (Canonical form of constant-rank maps): A smooth map $F$ of constant rank $r$
between smooth manifolds $M$ and $N$ of dimensions $m$ and $n$
has a local coordinate representation of the canonical form
$\hat{F}(x^i)_{i=1}^m = ((x^i)_{i=1}^r, 0_{n-r})$ at each point $p \in M$
w.r.t. some smooth coordinate charts $\phi$ and $\psi$ centered at $p$ and $F(p)$.
Corollary:
A smooth map $F$ on a connected manifold $M$ has local coordinate representations $\hat{F}$
which are linear maps and whose domains cover $M$, if and only if $F$ has constant rank.

*Global Rank Theorem*: For a smooth map $F: M \mapsto N$ of constant rank:
if it is surjective, then it is a smooth submersion;
if it is injective, then it is a smooth immersion;
if it is bijective, then it is a diffeomorphism.

**Smooth embedding** $F: M \mapsto N$ of a smooth manifold $M$ into another smooth manifold $N$
is a smooth immersion---not just a smooth map---that is also a topological embedding.
A smooth immersion may not be a topological embedding,
e.g. the curve $\beta: (-\pi, \pi) \mapsto \mathbb{R}^2$ defined by $\beta(t) = (\sin 2t, \sin t)$
maps an open interval to a lemniscate (8-shape).
A smooth map is a smooth immersion if and only if it equals some smooth embeddings glued together.
Although any smooth manifold admits a smooth embedding into some Euclidean space
(see the strong Whitney embedding theorem),
working with manifolds as abstract topological spaces can be an advantage,
both in complexity and interpretability, e.g. spacetime in general relativity.

... A smooth map $\pi: M \mapsto N$ is a smooth submersion if and only if the images of smooth local sections of $\pi$ cover $M$. Any smooth submersion is an open map; if it is surjective, it is a quotient map. A map $F: N \mapsto P$ between smooth manifolds is smooth if and only if the composition of any surjective smooth submersion $\pi: M \mapsto N$ onto $N$ is smooth. If a smooth map $F: M \mapsto P$ is constant on the level sets of a surjective smooth submersion $\pi: M \mapsto N$, then there exists a unique smooth map $\tilde{F}: N \mapsto P$ such that $F = \tilde{F} \circ \pi$.

**Smooth covering map** $\pi: E \mapsto M$ is a covering map that is also a local diffeomorphism.
Every smooth covering map is a smooth submersion, an open map, and a quotient map.

Smooth **submanifold** $(S, \mathcal{T}, \mathcal{A})$ of a smooth manifold $M$
is a subset of the manifold endowed with a topology and a smooth structure
such that it is a smooth manifold.
In the context of a submanifold of a smooth manifold,
we call the manifold the **ambient manifold** of the submanifold.
Although a submanifold is a manifold itself and thus can be treated in an abstract way,
we often consider it as a subset of the ambient manifold,
which provides a global representation of the submanifold
however the ambient manifold is locally represented in a Euclidean space.

**Immersed submanifold** of a smooth manifold is a smooth submanifold without boundary
such that the inclusion map $\iota: S \mapsto M$ is a smooth immersion.
Note that the topology of an immersed submanifold is not necessarily the subspace topology.
By convention, smooth submanifold refers to immersed submanifold.
**Embedded submanifold** of a smooth manifold is a smooth submanifold without boundary
such that the topology is the subspace topology and the inclusion map is a smooth embedding.
Immersed submanifolds are locally embedded.
**Properly embedded submanifold** of a smooth manifold is an embedded submanifold
such that the inclusion map is a proper map.
An embedded submanifold is properly embedded if and only if
it is a closed subset of the ambient manifold.
Every compact embedded submanifold is properly embedded.

**Codimension** (余维度) $\text{codim} S$ of an embedded submanifold of a smooth manifold
is the difference between their dimensions: $\text{codim} S = \dim M - \dim S$.
**Open submanifold** $(U, \mathcal{T}_U, \mathcal{A}_U)$ of a smooth manifold $M$
is an open subset of $M$ endowed with the subspace topology and the subspace smooth structure.
The open submanifolds of a smooth manifold are all its embedded submanifolds of codimension zero.
**Regular domain** in a smooth manifold $M$
is a properly embedded codimension-0 submanifold with boundary.
**Embedded hypersurface** is an embedded submanifold of codimension one.

The image $F(M_1)$ of any smooth embedding $F: M_1 \mapsto M_2$, endowed with the subspace topology of $\mathcal{T}_2$ and the smooth structure $\{\phi \circ F^{-1} : \phi \in \mathcal{A}_1\}$, is an embedded submanifold of $M_2$ diffeomorphic to $M_1$ via $F$; the said smooth structure is the only smooth structure such that $F$ is a diffeomorphism onto its image $F(M_1)$.

The graph $\Gamma(f)$ of any smooth map $f: U \times N$ from any open subset $U$ of any smooth n-manifold $M$ without boundary, endowed with the subspace topology and the smooth structure determined by the smooth atlas consisting of the graph coordinate chart, is a properly embedded n-submanifold of $M \times N$ diffeomorphic to the domain $U$ via $f$.

The most useful kinds of embedded submanifolds are those identified as level sets.
*Constant-Rank Level Set Theorem*: Any level set of a smooth map of constant rank $r$
is a properly embedded submanifold in the domain with codimension $r$.
(For a local, $C^k$ version, see [@Rudolph2013, Prop 1.7.6].)
*Submersion Level Set Theorem*: Any level set of a smooth submersion
is a properly embedded submanifold in the domain with codimension $\dim N$.
Embedded submanifolds are locally level sets of a smooth submersion:
A subset of a smooth manifold is an embedded submanifold with codimension $n$ if and only if
it has an open cover consisting of level sets of smooth submersions to $\mathbb{R}^n$.
**Regular point** and **critical point** $p$ of a smooth map $\Phi: M \mapsto N$
is a point in $M$ where the differential $d \Phi_p$ is surjective / not surjective.
**Regular value** and **critical value** $c$ of a smooth map
is a point in $N$ such that the level set $\Phi^{-1}(c)$ has no / some critical points.
**Regular level set** is a level set consisting of regular points.
*Regular Level Set Theorem*: Any regular level set of a smooth map
is a properly embedded submanifold in $M$ with codimension $\dim N$.
In other words, a level set of a smooth map $\Phi: M \mapsto N$
is a properly embedded (m-n)-submanifold in $M$
if $\Phi$ is surjective, i.e. has rank $n$, on the level set.
*Regular Level Set Theorem* ($C^k$ version) [@Hirsch1976, Thm 3.2]:
Any regular level set of a $C^k$ map between $C^k$ manifolds, $k \ge 1$,
is a properly embedded $C^k$ submanifold in the domain with codimension $\dim N$.
**Defining map** for an embedded submanifold $S$ of a smooth manifold $M$
is a smooth map $\Phi$ on $M$ such that $S$ is a regular level set.
**Defining function** is a defining map whose codomain is a Euclidean space.

**Smooth local parameterization** $X: \phi(U) \mapsto M$ of a submanifold of a smooth manifold
is the inverse of a smooth chart of the submanifold followed by the inclusion map:
$X = \iota \circ \phi^{-1}$, where $\iota: S \mapsto M$, $\phi: U \mapsto \mathbb{R}^n$.
The domain of a smooth local parameterization of an n-submanifold is an open subset of
the Euclidean n-space, its closed upper half-space, or its nonnegative space,
if the submanifold is without boundary, with boundary, or with corners, respectively.
Smooth local parameterizations of an n-submanifold of the Euclidean m-space is a function
from a subset of the Euclidean n-space into the Euclidean m-space.
**Graph parameterization** of the graph of a smooth real-valued function
$f: U \mapsto \mathbb{R}$ on an open subset of the Euclidean n-space
is the smooth global parameterization $\Phi_f: U \mapsto \mathbb{R}^{n+1}$, $\Phi_f(x) = (x, f(x))$.

**Measure zero** subset $A$ in a smooth n-manifold $M$
is a subset whose image has zero n-volume in every smooth chart:
$\forall (U, \phi) \in \mathcal{A}$, $\lambda(\phi(A \cap U)) = 0$.
*Sard’s theorem*:
The set of critical values of a smooth map $F: M \mapsto N$ has measure zero in $N$.

*Whitney embedding theorem* [@Whitney1936]:
Any smooth n-manifold admits a proper smooth embedding into $\mathbb{R}^{2n+1}$.
Note that 2n comes from the dimensions of
the product manifold $M \times M$ and the tangent bundle $T M$.
The Whitney embedding theorem imply that the intrinsic/metric definition of smooth manifolds
is no more general than the extrinsic/submanifold definition.
*Strong Whitney embedding theorem* [@Whitney1944a]:
Any smooth n-manifold, $n > 0$, admits a smooth embedding into $\mathbb{R}^{2n}$.
This is the best possible embedding dimension for smooth 1- and 2-manifolds,
e.g. Klein bottle is a smooth 2-manifold that always self-intersects in $\mathbb{R}^3$.
However, every smooth 3-manifold can be embedded in $\mathbb{R}^5$ [@Wall1965].

*Whitney immersion theorem*:
Any smooth n-manifold admits a smooth immersion into $\mathbb{R}^{2n}$.
*Strong Whitney immersion theorem*: [@Whitney1944b]
Any smooth n-manifold, $n > 1$, admits a smooth immersion into $\mathbb{R}^{2n-1}$.
The best possible immersion dimension is provd by [@Cohen1985]:
Every compact smooth n-manifold can be immersed in $\mathbb{R}^{2n-a(n)}$,
where $a(n)$ is the number of 1’s in the binary expression for $n$.
For example, every 3-manifold can be immersed in $\mathbb{R}^4$,
every 4-manifold can be immersed in $\mathbb{R}^7$.

**Normal space** $N_p M$ at a point $p$ on an embedded m-submanifold $M$ of $\mathbb{R}^n$
is the (n-m)-dimensional subspace of the tangent space $T_p \mathbb{R}^n$
that consists of all vectors orthogonal to the tangent space $T_p M$
with respect to the Euclidean inner product.
**Normal bundle** (法丛) $N M$ of an embedded submanifold $M$ of $\mathbb{R}^n$
is the disjoint union of the normal spaces at all points of $M$:
$N M = \sqcup_{p \in M} N_p M$.
The normal bundle of any embedded submanifold of $\mathbb{R}^n$
is an embedded n-submanifold of the tangent bundle $T \mathbb{R}^n$.
**Normal exponential map** $E: \mathscr{E}_M \mapsto \tilde M$
of an embedded submanifold in a Riemannian manifold
is the restriction of the exponential map of the ambient manifold
to (the intersection of its domain with) the normal bundle of the submanifold:
$E = \exp|_{\mathscr{E}_M}$, where $\mathscr{E}_M = \mathscr{E} \cap N M$.
In case the ambient manifold is Euclidean, the normal exponential map
can be identified with the addition map: $E(p, v) = p + v$.
**Normal neighborhood** $U$ of an embedded submanifold $M$ of $\mathbb{R}^n$
is a neighborhood that is diffeomorphic, via the addition map,
to a fiber bundle of star-shaped neighborhoods of the origin in the normal spaces:
$U = E(V)$, $E(p, v) = p + v$, $V = \{(p, v) \in NM: v \in S_p\}$,
where $S_p$ is a star-shaped neighborhood of the origin in $N_p M$.
**Tubular neighborhood** $U$ of an embedded submanifold $M$ of $\mathbb{R}^n$
is a neighborhood that is diffeomorphic, via the addition map,
to a fiber bundle of normal disks of continuous radius: $U = E(V)$, $E(p, v) = p + v$,
$V = \{(p, v) \in NM: |v| < \delta(p)\}$, $\delta \in C^0(M, \mathbb{R}_+)$.
**ε-tubular neighborhood** or **uniform tubular neighborhood** of radius ε
is a tubular neighborhood with constant radius: $\delta(p) = \varepsilon$.
Any embedded submanifold of $\mathbb{R}^n$ has a tubular neighborhood.
**Retraction** $r: X \mapsto M$ of a topological space $X$ onto a subspace $M$
is a surjective continuous map whose restriction to the codomain is the identity map:
$r|_M = \text{Id}_M$.
Any tubular neighborhood $U$ of an embedded submanifold $M$ of $\mathbb{R}^n$
has a retraction $r: U \mapsto M$ that is also a smooth submersion.
**Fermi coordinates** $\phi: U \mapsto \mathbb{R^n}$ on a normal neighborhood
of an embedded d-submanifold of a Riemannian n-manifold
is a coordinate map that concatenates a local coordinate chart of the submanifold
and a local orthonormal frame for the normal bundle [@Gray1982; @Gray2004]:
$\phi(E(p, v)) = (x^i, v^j)_{i \in d, j \in n-d}$.

*Whitney Approximation Theorem* for functions [@Whitney1936]:
Any continuous function on a smooth manifold can be uniformly approximated by a smooth function:
$F \in C^0(M, \mathbb{R}^k)$, $F|_A \in C^\infty, A \in \mathcal{T}^∗$,
$\forall \delta \in C^0(M, \mathbb{R}_+)$, $\exists \tilde{F} \in C^\infty(M, \mathbb{R}^k)$,
$|\tilde{F} - F| < \delta$, $\tilde{F}|_A = F|_A$.
*Whitney Approximation Theorem* [@Whitney1936]:
Any continuous map between smooth manifolds is homotopic to a smooth map;
if the map is smooth on a closed subset of the domain and the codomain has no boundary,
then the homotopy can be taken to be relative to the subset:
$\forall F \in C^0(M, N)$, ($F|_A \in C^\infty, A \in \mathcal{T}^∗$, $\partial N = \emptyset$),
$\exists \tilde{F} \in C^\infty(M, N)$, $\tilde{F} \simeq F$, ($\tilde{F}|_A = F|_A$).
*Extension Lemma for Smooth Maps*:
A smooth map $f: A \mapsto N$ from a closed subset $A$ of a smooth manifold $M$
to a smooth manifold without boundary can be extended to a smooth function $\tilde{f}$ on $M$
if and only if it has a continuous extension on $M$.
If two smooth maps are homotopic, then they are smoothly homotopic;
if they are homotopic relative to a closed subset of the domain and the codomain has no boundary,
then they are smoothly homotopic relative to the subset.

Consistently oriented bases of a finite-dimensional vector space... Orientation for a finite-dimensional vector space. Oriented vector space... A vector space has exactly two choices of orientation. Pointwise orientation on a smooth manifold... Oriented frame... Orientation on a smooth manifold is a continuous point-wise orientation, i.e. ... Orientable manifold... Nonorientable manifold. A orientable connected smooth manifold has exactly two orientations. Two orientations of a connected, orientable, smooth manifold are equal if they agree at one point. Oriented manifold... Orientation form... Oriented form... Oriented smooth chart... Consistently oriented smooth altas...

Fibration (aka fiber space, fiber bundle) is a way to decompose a high-dimensional manifold into a manifold-indexed collection of homeomorphic low-dimensional manifolds, endowed with a manifold topology and a smooth structure such that it is locally homeomorphic to (subpaces of) product spaces.

Commutative diagram of fiber bundle.

**Fiber bundle** $(X, \pi, \{\Phi_\alpha\}_{\alpha \in A})$ over a topological space $B$
is a topological space $X$ endowed with a quotient map $\pi: X \mapsto B$
and a collection of homeomorphisms $\Phi_\alpha: \pi^{-1}(U_\alpha) \mapsto U_\alpha \times F$
from saturated subsets to product spaces,
whose domains form an open cover and whose first component maps match the quotient map:
$\cup_\alpha U_\alpha = B$, $\cup_\alpha \Phi^1_\alpha = \pi$.
We call $X$ the **total space** of the bundle, $B$ its **base space**,
$\pi$ its canonical **projection**, topological space $F$ its **model fiber**,
and $\Phi$ a **local trivialization** of $X$ over $U$.
Denote $X_p$ the fiber over a point $p$ in the base space: $X_p = \pi^{-1}(p)$.
**Global trivialization** of a fiber bundle is a local trivialization over its base space.
**Product fiber bundle** $(B \times F, \pi_1)$ is a fiber bundle consisting of
the product space of the base space and its model fiber, and the first canonical projection.
**Trivial fiber bundle** is a fiber bundle that admits a global trivialization.
Equivalently, a trivial fiber bundle is one
that is homeomorphic to the product fiber bundle of its base space and its model fiber:
$X \cong B \times F$.
**Smooth fiber bundle** is a fiber bundle
where $X, B, F$ are smooth manifolds, $\pi$ is a smooth map,
and the local trivializations can be chosen to be diffeomorphisms.
**Smooth local trivialization** on a smooth fiber bundle
is a local trivialization that is a diffeomorphism onto its image.
**Smoothly trivial fiber bundle** is a smooth fiber bundle that admits a smooth global trivialization.
Equivalently, a smoothly trivial fiber bundle is one
that is diffeomorphic to the product fiber bundle $B \times F$.

Real **vector bundle** $(E, \pi, \{\Phi_\alpha\}_{\alpha \in A})$ of rank $k$ over $B$
is a fiber bundle with model fiber $\mathbb{R}^k$,
such that every restriction $\Phi_\alpha |_{E_p}$ of a local trivialization to a fiber
is a vector space isomorphism to $\{p\} \times \mathbb{R}^k$.
Real **line bundle** is a rank-1 vector bundle.
**Smooth vector bundle** is a vector bundle that is a smooth fiber bundle.
The Möbius bundle is a smooth line bundle over the circle that is not trivial.
A smooth vector bundle is smoothly trivial if and only if it admits a smooth global frame.
**Transition function** $\tau: U_1 \cap U_2 \mapsto \text{GL}(k, \mathbb{R})$
between smooth local trivializations on a smooth rank-$k$ vector bundle
is a smooth map to invertible matrices,
such that the composite map $\Phi_2 \circ \Phi_1^{-1}$,
which is a transformation on $(U_1 \cap U_2) \times \mathbb{R}^k$,
can be written as $\Phi_2 \circ \Phi_1^{-1} (p, v) = (p, \tau(p) v)$.

The tangent bundle of a smooth n-manifold together with its natural projection map, $(T M, \pi)$, is a smooth vector bundle of rank $n$ over the manifold. For a tangent bundle, each transition function between local trivializations associated with two smooth charts equals the Jacobian matrix of the transition map between these charts.

*Vector Bundle Construction Theorem* (construction by surgery):
Given an open cover $U_\alpha$ of a smooth manifold $M$ and a collection of smooth maps
$\tau_{\alpha\beta}: U_\alpha \cap U_\beta \mapsto \text{GL}(k, \mathbb{R})$
that are transitive $\tau_{\alpha\beta} \tau_{\beta\gamma} = \tau_{\alpha\gamma}$,
one can construct a smooth rank-$k$ vector bundle
$((\sqcup_p E_p, \mathcal{T}_\tau, \mathcal{A}_\tau), \pi, \{\Phi_\alpha\}_{\alpha \in A})$
over $M$ such that the transition functions among its smooth local trivializations are
$\tau_{\alpha\beta}$.

**Restriction** $X|_S$ of a fiber bundle to a subset $S$ of its base space
is the fiber bundle $(\pi^{-1}(S), \pi|_S, \{\Phi_\alpha|_S\}_{\alpha \in A})$,
where $\pi^{-1}(S)$ is a subspace of $X$.
The restriction $E|_S$ of a smooth vector bundle to a smooth submanifold of its base manifold,
together with its subspace smooth structure, is a smooth vector bundle.
**Ambient tangent bundle** $T M|_S$ of a smooth submanifold $S$ over its ambient manifold $M$
is the restriction of the tangent bundle of the ambient manifold to the submanifold.

**Subbundle** $(D, \pi|_D, \{\Phi_\alpha|_D\}_{\alpha \in A})$ of a vector bundle $E$
is a vector bundle where $D$ is a topological subspace of $E$
such that $D_p$ is a linear subspace of $E_p$.

**Whitney sum** $E' \oplus E''$ of smooth rank-$k'$ and rank-$k''$ vector bundles
over a smooth manifold $M$ is the rank-$(k'+k'')$ vector bundle over $M$
determined by transition functions $\tau = \tau' \oplus \tau''$
(where $\oplus$ is the direct sum of matrices):
$((\sqcup_p (E'_p \oplus E''_p), \mathcal{T}_\tau, \mathcal{A}_\tau), \pi)$,
where $\oplus$ is the direct sum of vector spaces.

Global **section** or **cross section** $\sigma$ of a vector bundle
is a section of its canonical projection: $\sigma \in C(B, E)$, $\pi \circ \sigma = \text{Id}_B$.
**Local section** of a vector bundle is a local section of its canonical projection:
$\sigma \in C(U, E)$, $U \subset B$, $\pi \circ \sigma = \text{Id}_U$.
**Smooth section** of a vector bundle is a section that is a smooth map.
The set $\Gamma(E)$ of all smooth (global) sections of a smooth vector bundle
is a vector space under pointwise addition and scalar multiplication.
**Rough section** of a vector bundle
is a right inverse of the canonical projection over a subset of the base space:
$\sigma: U \mapsto E$, $U \subset B$, $\pi \circ \sigma = \text{Id}_U$.
**Zero section** $\zeta$ of a vector bundle is the global section
that takes every point in the base space to the zero vector at that point:
$\forall p \in B$, $\zeta(p) = 0 \in E_p$.
**Support** of a section is the closure of its non-zero set: $\overline{B \setminus \sigma^{-1}(0)}$.

**Local frame** $(E_i)_{i=1}^n$ for a smooth n-manifold is an ordered n-tuple of vector fields
on an open subset of the manifold, which is linearly independent and spans the vector bundle,
i.e. it is a basis for the fiber space at each point.
**Global frame** is a frame defined on the entire manifold.
**Smooth frame** is a frame consisting of smooth vector fields.

**Fiber-space morphism** from a fiber space $\pi_1: X_1 \mapsto B_1$
into another $\pi_2: X_2 \mapsto B_2$ is a map $\phi: X_1 \mapsto X_2$
such that $\pi_2 \circ F \circ \pi_1^{-1}$ is a map $f: B_1 \mapsto B_2$.
**Fiber-space isomorphism** is a morphism between $\pi_1$ and $\pi_2$
that is a homeomorphism (topological isomorphism) between $X_1$ and $X_2$.

**Bundle homomorphism** (丛同态) $F: E \mapsto E'$ between vector bundles
is a continuous map that is linear on each fiber
and there is a map $f: B \mapsto B'$ between the base spaces
such that $\pi' \circ F = f \circ \pi$.
The bundle homomorphism $F$ is said to **cover** the map $f$ between the base spaces.
The map covered by a bundle homomorphism is continuous and unique.
The map covered by a smooth bundle homomorphism between smooth vector bundles is smooth.
**Bundle isomorphism** (丛同构) is a bijective bundle homomorphism
whose inverse is also a bundle homomorphism.
**Smooth bundle isomorphism** is a bundle isomorphism that is a diffeomorphism.

**Bundle homomorphism** between vector bundles over the same base space
is bundle homomorphism that covers the identity map of the base space.
Any bijective smooth bundle homomorphism over a smooth manifold is a smooth bundle isomorphism.

Algebraic topology is the subject that studies topological properties of topological spaces by attaching algebraic structures to them in a topologically invariant way, see Abstract Algebra.

Complex (复形). Cell complex (胞腔复形). CW complex. Chain complex (链复形). Cochain complex (上链复形).

**Simplicial complex** (单纯复形) $K$ is a space with a triangulation:
$K = \{s_i\}_i \subset \mathbb{R}^n$ is a class of simplices
such that every face of a simplex $s_i$ is in $K$,
and the intersection of any two simplices is a face of each of them.
Graph (in discrete math) as a topological space is equivalent to simplicial 1-complex.

**Topological group** $(G, ∗, \mathcal{T})$ is a group with a topology
such that the group operation and the inverse map are continuous:
$∗ \in C(G^2, G)$, $∗^{-1}(e) \in C(G, G)$.
**Lie group** $(G, ∗, \mathcal{T}, \mathcal{A})$ is a smooth manifold with a group structure
such that the multiplication and inversion maps are smooth maps.
**Matrix Lie group** is a subgroup of the complex general linear group $\text{GL}_n(\mathbb{C})$,
e.g. the real general linear group $\text{GL}_n(\mathbb{R})$ and the orthogonal group O(n).
Every element in a matrix Lie group, as a left action, is a diffeomorphism:
$\forall A \in G$, let $L_A(B) = A B$, then $L_A: G \cong G$.
**Representation** $\rho: G \mapsto \text{GL}(V)$ of a finite-dimensional Lie group
is a Lie group homomorphism to the general linear group on a finite-dimensional vector space.
**Faithful representation** is an injective representation.
Exponential map $\exp: \mathfrak{g} \mapsto G$.

**Lie bracket** $[X, Y]$ of a pair of smooth vector fields on a smooth manifold
is the smooth vector field such that applying it to any smooth function equals the difference
in the functions obtained by applying the pair to the function in different orders:
$\forall f \in C^\infty(M)$, $[X, Y] f = X Y f - Y X f$.
The Lie bracket of any pair of smooth vector fields is a smooth vector field:
$\forall X, Y \in \mathfrak{X}(M)$, $[X, Y] \in \mathfrak{X}(M)$.
The Lie bracket of a pair of smooth vector fields has coordinate representation:
$[X, Y] = \left(X^i \frac{\partial Y^j}{\partial x^i} - Y^i \frac{\partial X^j}{\partial x^i}\right)
\frac{\partial}{\partial x^j}$.
The Lie bracket operator on smooth vector fields is bilinear, anti-symmetric, and satisfies:
(1) Jacobi identity: $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$;
(2) $[f X, g Y] = f g [X, Y] + (f X g) Y - (g Y f) X$, where $f, g \in C^\infty(M)$.
**Lie algebra** $(\mathfrak{g}, (+, \cdot_{\mathbb{R}}), [\cdot, \cdot])$ over real numbers
is a real vector space endowed with a map (called the **bracket**)
that is bilinear, anti-symmetric, and satisfies the Jacobi identity.
**Lie algebra** $\text{Lie}(G)$ of a Lie group G
is the Lie algebra of all left-invariant vector fields on the Lie group.
The Lie algebra $(T_I G, (+, \cdot_{\mathbb{R}}), [\cdot, \cdot])$ of a matrix Lie group G
is its tangent space at the identity endowed with the **matrix commutator** as the bracket:
$[V, W] = V W - W V$, where $V, W \in T_I G$.
Note that the tangent space of a matrix Lie group G at an element A can be written as:
$T_A G = A \text{Lie}(G) = \{A V \in M_n : V \in T_I G\}$.
The Lie algebra $\mathfrak{so}(n)$ of the orthogonal group
consists of the set of skew-symmetric matrices:
$\mathfrak{so}(n) := \text{Lie}(O(n)) = T_I O(n) = \Omega(n)$,
where $\Omega(n) = \{\Omega \in M_n : \Omega = - \Omega^T \}$.

**Lie derivative** $\mathscr{L}_V W$ of $W$ w.r.t. $V$,
both smooth vector fields on a smooth manifold, is the rough vector field on the manifold defined by:
$(\mathscr{L}_V W)_p = \frac{d}{d t}\bigg|_{t=0} d(\theta_{-t})_{\theta_t(p)}(W_{\theta_t(p)})$,
i.e. $(\mathscr{L}_V W)_p = \lim_{t \to 0} (d(\theta_{-t})_{\theta_t(p)}(W_{\theta_t(p)}) -
W_p) / t$, where $\theta$ the flow of $V$.
Lie derivative equals Lie bracket: $\forall V, W \in \mathfrak{X}(M)$, $\mathscr{L}_V W = [V, W]$.

*Closed subgroup theorem*:
Every topologically closed subgroup of a Lie group is actually an embedded Lie subgroup.
There is a one-to-one correspondence between
isomorphism classes of finite-dimensional Lie algebras and
isomorphism classes of simply connected Lie groups.

Left **action** of a topological group G on a topological space X
is a map $\phi : G \times X \mapsto X$ such that:
$\forall g_1, g_2 \in G$, $\forall p \in X$,
$\phi(g_1, \phi(g_2, p)) = \phi(g_1 g_2, p)$ and $\phi(e, p) = p$ (e is the identity element of G).
If the action is unambiguous, one may simply write: $\phi(g, p) = g \cdot p$;
in comparison, the group operation is omitted altogether: $g_1 ∗ g_2 = g_1 g_2$.
Note that since the map takes two variables of distinct types,
for definiteness, one usually put the group element first, i.e. the group acts on the left;
similarly, a **right action** is an action where the group acts on the right.
**Orbit** $G \cdot p$ of a point $p \in X$ is the image of the group acting on the point:
$G \cdot p = \{g \cdot p : g \in G\}$.
**Orbit space** $X / G$ of an action is the quotient space of orbits:
$X / G = \{G \cdot p : p \in X\}$;
here the quotient map $\pi: X \mapsto X / G$ takes each point to its orbit $\pi(p) = G \cdot p$.
**Isotropy group**, **stationary subgroup**, or **stabilizer** $G_p$
of a point p in a set X with action G
is the subgroup where the point is a fixed point:
$G_p = \{g \in G : g \cdot p = p\}$.

**Smooth action** of a Lie group on a smooth manifold is an action that is a smooth map.
**Free action** is one such that every non-identity element in the group acts without fixed points:
if $\exists p \in M$, g · p = p, then g = e;
or equivalently, every isotropy group is trivial: $\forall p \in M$, $G_p = \{e\}$.
**Proper action** is one such that each compact subset is moved away from itself
by most elements of the group.
Every continuous action by a compact Lie group on a manifold is proper.
*Quotient manifold theorem*:
If a Lie group G acts smoothly, freely, and properly on a smooth manifold M,
then the orbit space M/G is a smooth manifold of dimension dim(M) - dim(G),
whose smooth structure is uniquely determined such that
the quotient map $\pi: M \mapsto M / G$ is a smooth submersion.
In this case, we call the orbit space M/G a **quotient manifold**.

**Equivariant map** between smooth manifolds with smooth G-actions
is a map that commutes with the action:
$\forall g \in G$, $F(g \cdot p) = g \cdot F(p)$;
order is switch for right actions.
*Equivariant rank theorem*:
Let M and N be smooth manifolds with smooth G-actions.
If the G-action on M is transitive (i.e. (M, G) is a homogeneous G-space),
then any smooth equivariant map $F: M \mapsto N$ constant rank.
(Therefore the global rank theorem applies.)

*Orbits of proper actions*:
Given a proper smooth action ϕ of a Lie group G on a smooth manifold M,
the orbit map $\phi^{(p)}: G \mapsto M$ of any point p ∈ M is a proper map,
and thus the orbit G·p is closed in M.
If its isotropy group $G_p$ contains only the identity element,
then the orbit map is a smooth embedding,
and the orbit is a properly embedded submanifold.
*Theorem*:
If a compact Lie group G acts smoothly on a smooth manifold M,
then the orbit G·p of any point p ∈ M is a properly embedded submanifold of M,
and it is diffeomorphic to the quotient manifold $G / G_p = \{G_p g : g \in G\}$.
Note that $G / G_p$ is a quotient manifold
because the isotropy group acts smoothly and freely on the compact Lie group,
where the action is the group operation.

Every Lie group admits a left-invariant smooth global frame.

**Homogeneous G-space**, **homogeneous space**, or **homogeneous manifold** (M, G)
is a smooth manifold M with a smooth transitive action by a Lie group G:
(1) $\forall x, y \in M, \exists g \in G: g \cdot x = y$ (transitivity);
(2) $e \cdot x = x$ (identity map);
(3) $(g h) \cdot x = g \cdot (h \cdot x)$ (composition);
The elements of G are called the **symmetries** of M.
The isotropy group $G_p$ of any point p in a homogeneous G-space is a closed subgroup of G.
**Left coset** $g H$ of a Lie subgroup H of a Lie group G
is the subset $g H = \{g h : h \in H\}$, where $g \in G$.
**Left coset space** of G modulo H, denoted by $G / H$,
is the quotient space of all left cosets of H, that is, $G / H = \{g H : g \in G\}$.
We say two elements $g_1, g_2 \in G$ are **congruent modulo H**, $g_1 \equiv g_2 (\text{mod} H)$,
if they are in the same left coset of H, or equivalently $g_1^{-1} g_2 \in H$
(here $g^{-1}$ denotes an inverse element).
*Homogeneous space construction theorem*:
Given any closed subgroup H of G, the left coset space (G / H, G) is a homogeneous space.
*Homogeneous space characterization theorem*:
For every homogeneous space (M, G), let p be an arbitrary point in M,
then (M, G) can be identified with the left coset space $(G / G_p, G)$,
and $F: G / G_p \mapsto M$ defined by $F(g G_p) = g \cdot p$ is an equivariant diffeomorphism.

**Reductive (homogeneous) space** is a homogeneous space G / H of a connected Lie group G
such that there is a subspace of $\mathfrak{g}$
that is invariant under $\text{Ad}_{\mathfrak{g}}(H)$ and complementary to $\mathfrak{h}$.
Here, $\mathfrak{g}$ and $\mathfrak{h}$ are the Lie algebras of G and H respectively,
and $\text{Ad}_{\mathfrak{g}}$ is the adjoint representation of $\mathfrak{g}$.
Every homogeneous Riemannian space is reductive.

Examples. General linear group $\text{GL}(n, \mathbb{F})$ is a Lie group under matrix multiplication. The group $\text{GL}(V)$ of vector space isomorphisms on any n-dimensional vector space over the field $\mathbb{F}$ is a Lie group isomorphic to the general linear group $\text{GL}(n, \mathbb{F})$. Special linear group $\text{SL}(n, \mathbb{F})$ is a Lie subgroup of the general linear group. Orthogonal group $O(n)$ / unitary group $U(n)$ / special orthogonal group $SO(n)$ / special unitary group $SU(n)$ is a Lie subgroup of the real / complex, general / special linear group.

**Direct product group** $\prod_{i=1}^k G_i$ of Lie groups
is the Lie group consisting of the product manifold and the component-wise group multiplication:
$(g_i)_{i=1}^k (g_i')_{i=1}^k = (g_i g_i')_{i=1}^k$.
**Semidirect product group** $N \rtimes_\theta H$
of Lie groups determined by a smooth left action by automorphisms $\theta: H \times N \mapsto N$
is the Lie group consisting of the product manifold and the group multiplication defined by
$(n, h)(n', h') = (n (h n'), h h')$.

**Homotopy** (同伦) $H: X \times I \mapsto Y$, $I = [0, 1]$,
between two continuous maps $f, g: X \mapsto Y$ between the same two topological spaces
is a continuous map that matches the two maps on the boundaries:
$H(x, 0) = f(x)$ and $H(x, 1) = g(x)$; denoted as $H: f \simeq g$.
Two continuous maps $f, g: X \mapsto Y$ are **homotopic**, denoted as $f \simeq g$,
if there is a homotopy between them.
For any pair of topological spaces $X$ and $Y$,
homotopy is an equivalence relation on the set $C^0(X, Y)$ of all continuous maps between them;
the set of homotopy classes of continuous maps from $X$ to $Y$ is denoted by $[X, Y]$.
**Homotopy inverse** $\psi: Y \mapsto X$ for a continuous map $\phi: X \mapsto Y$
is a continuous map such that their compositions are homotopic to identity maps:
$\psi \circ \phi \simeq \text{Id}_X$ and $\phi \circ \psi \simeq \text{Id}_Y$.
**Homotopy equivalence** $\phi: X \mapsto Y$ is a continuous map that has a homotopy inverse.
Two topological spaces $X$ and $Y$ are **homotopy equivalent**, denoted as $X \simeq Y$,
if there is a homotopy equivalence between them.
Homotopy equivalence is an equivalence relation on the class of all topological spaces.
**Homotopy invariant** of a homotopy equivalent class of topological spaces
is a property common to all these spaces.
Any homeomorphism is a homotopy equivalence, and thus
topological equivalence is a finer equivalence relation than homotopy equivalence, and
homotopy invariants are topological invariants.
Examples of homotopy invariants: Euler characteristic, fundamental groups,
singular homology groups, De Rham cohomology groups, homotopy groups.

**Smooth homotopy** between two smooth maps between the same two smooth manifolds
is a homotopy that is also a smooth map, in the sense that
it extends to a smooth map on some neighborhood of $X \times I$ in $X \times \mathbb{R}$.
**Isotopy** (同痕) $H: X \times I \mapsto Y$ between two embeddings of $X$ into $Y$
is a homotopy such that for all $t \in (0, 1)$, $H(x, t)$ is an embedding.

**Homotopy relative to**, or **stationary** on, a subset of the domain of the related maps
is one such that at any time its restriction to the subset is the same:
$\forall x \in A \subset X$, $\forall t \in I$, $H(x, t) = H(x, 0)$.
Two continuous maps matching on a subset of their domain are **homotopic relative to** the subset
if there is a homotopy between them relative to the subset.
Given a homotopy that is not stationary on any subset,
we say the two homotopic maps are **freely homotopic**.
**Path homotopy** $H: I \times I \mapsto X$ between two paths
$f, g: I \mapsto X$ sharing the same endpoints in a topological space
is a homotopy that is stationary at the endpoints:
$H(0, t) = f(0) = g(0)$ and $H(1, t) = f(1) = g(0)$; denoted as $H: f \sim g$.
Two paths sharing the same endpoints are **path-homotopic**, denoted as $f \sim g$,
if there is a path homotopy between them.

**Fundamental group** $\pi_1(X, p)$ of a topological space $X$ based at a point $p$
is the set of path classes of loops based at $p$.
Two paths $f, g: I \mapsto X$ are **composable** if they match in tandem: $f(1) = g(0)$.
**Product** $f \cdot g$ of two composable paths $f$ and $g$ is the path connecting them in tandem:
$f \cdot g (s) = f(2s), s \in [0,1/2]; g(2s-1), s \in [1/2, 1]$.
The fundamental group $\pi_1(X, p)$ of any topological space $X$ at any point $p \in X$,
together with path product $\cdot$, is a group.
The fundamental group, in a certain sense, measures the number of holes in a topological space.
Homeomorphic spaces have isomorphic fundamental groups.
Homotopy equivalent spaces have isomorphic fundamental groups.

*Homotopy Lifting Property*:
The lift of a homotopy $H: Y \times I \mapsto X$ between continuous maps
from a locally connected space, given a covering map $q: E \mapsto X$,
is uniquely determined by its "initial values":
$\tilde H_1(x, 0) = \tilde H_2(x, 0)$ then $\tilde H_1 = \tilde H_2$.
If the homotopy is stationary on some subset of its domain, then so are its lifts.
*Path Lifting Property*:
The lift of a path $f: I \mapsto X$, given a covering map $q: E \mapsto X$,
is uniquely determined by its initial value:
$\tilde f_1(0) = \tilde f_2(0)$ then $\tilde f_1 = \tilde f_2$.
*Monodromy Theorem*:
Given two paths with the same endpoints and a covering map,
their lifts with the same initial value are path-homotopic if and only if they are path-homotopic.

Two embedded submanifolds of a smooth manifold **intersect transversely** (横截交)
if their tangent spaces span the full tangent space whereever they intersect:
$S \pitchfork S'$ if $\forall p \in S \cap S'$, $\text{Span} (T_p S \cup T_p S') = T_p M$.
A smooth map is **transverse** (横截) to an embedded submanifold of its codomain
if the pushforward of the domain tangent spaces by the map and the submanifold tangent spaces
span the codomain tangent spaces whereever they intersect:
$F: N \mapsto M$, $S \subset M$, $F \pitchfork S$ if $\forall p \in F^{-1}(S)$,
$\text{Span} (d F_p(T_p N) \cup T_{F(p)} S) = T_{F(p)} M$.
Two embedded submanifolds intersect transversely
if and only if the inclusion of either one is transverse to the other.
(A generalization of the regular level set theorem:)
If two embedded submanifolds intersect transversely, then their intersection
is an embedded submanifold whose codimension is the sum of those of the two submanifolds.
*Transversality homotopy theorem*:
Given an embedded submanifold $S$ in $M$, every smooth map $f: N \mapsto M$
is homotopic to a smooth map $g: N \mapsto M$ that is transverse to $S$.

**Simply connected topological space**
is a path-connected topological space such that the fundamental group based at some point
is the trivial group consisting of only the path class of the constant path at the point.
For example, a sphere is simply connected, but a torus is not.
Simple connectivity is a topological invariant.

Functor. Fundamental group is a functor.

There are a number of functorial ways of associating to each topological space an algebraic object such as a group or a vector space, so that homeomorphic spaces have isomorphic objects. Most of these measure the existence of “holes” in different dimensions in a certain sense.

Homology (同调), cohomology (上同调). Simplicial homology (by H. Poincaré). Singular homology (by O. Veblen). Spectral homology (P.S. Aleksandrov).

De Rham cohomology group. De Rham groups are homotopy invariants, and thus topological invariants and diffeomorphism invariants. Mayer–Vietoris theorem. Degree of a continuous map between connected, compact, oriented, smooth n-manifolds.

A free abelian group has **finite rank** if it has a finite basis;
otherwise, we say it has **infinite rank.**
**Rank** of a free abelian group of a finite rank is the number of elements in any finite basis.

**Singular homology group** $H_p(X)$, $p \in \mathbb{N}$, is the quotient group
$H_p (X) = Z_p(X) / B_p(X)$, i.e. $H_p (X) = \text{Ker}\partial_p / \text{Im}\partial_{p+1}$.
A singular homology group is an Abelian group
which partially counts the number of holes in a topological space.
Singular cohomology groups.
Singular homology can be computed by restricting attention only to smooth simplices.
*de Rham Theorem*: Integration of differential forms over smooth simplices
induces isomorphisms between the de Rham groups and the singular cohomology groups.

**Betti number** $\beta_p(X)$ of a topological space, $p \in \mathbb{N}$,
is the rank of its p-th singular homology group: $\beta_p(X) = \text{rank}~H_p(X)$.
**Euler characteristic** $\chi(X)$ of a topological space is the alternating sum of its Betti numbers:
$\chi(X) = \sum_{p \in \mathbb{N}} (-1)^p \beta_p(X)$.
The Euler characteristic is a homotopy invariant.

Functorial classification of manifolds: The union of small balls around data points on the manifold $\hat{M} = \cup_i B(X_i, \varepsilon)$ has the same homology as the manifold $M$ with high probability, as long as $M$ has positive reach and $\varepsilon$ is small relative to the reach [@Niyogi2008].