Topology studies the invariant properties of objects under continuous transformations, e.g. stretch, twist; in comparison, geometry is constrained to transformations such as translation, rotation, and scaling.

General topology (or point-set topology, analytic toplogy) is the study of the general abstract nature of continuity on spaces, with fundamental concepts such as continuity, compactness, and connectedness. Topology can be further divided into algebraic topology (including combinatorial topology), differential topology, and low-dimensional topology.

## Concepts

Base (or topological basis) $\mathcal{B}$ of a topology on a set $X$ is a class $\mathcal{B} \subset \mathcal{P}(X)$ of its subsets such that: (1) $\mathcal{B}$ is a cover of $X$, i.e. $\cup_{B_\alpha \in \mathcal{B}} B_\alpha = X$; and (2) $\forall x \in B_1 \cap B_2$, $B_1, B_2 \in \mathcal{B}$, $\exists B_3 \in \mathcal{B}$: $x \in B_3 \subset B_1 \cap B_2$.

Topological structure $(\mathcal{T}, (\cup_\alpha, \cap))$, topology, or (in default) open topology of a set $X$ is a class $\mathcal{T} \subset \mathcal{P}(X)$ of its subsets which contains the empty set $∅$ and the full set $X$ and is closed in arbitrary union $\cup_\alpha$ and finite intersection $\cap$. Closed topology is the dual concept of open topology by exchanging 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 of a set. A topology $\mathcal{T}_1$ is said to be weaker than another topology $\mathcal{T}_2$, and similarly $\mathcal{T}_2$ stronger than $\mathcal{T}_1$, if the former specifies a coarser structure of the underlying set: $\mathcal{T}_1 \subset \mathcal{T}_2$. For a set $X$, the weakest topology is the class $\{\emptyset, X\}$ of the empty set and itself. the strongest topology is its power set $\mathcal{P}(X)$, aka the discrete topology.

Topology generated by a base $\mathcal{B}$ is the topology $\mathcal{T(B)}$ consisting of unions of members of arbitrary subsets of the base: $\mathcal{T(B)} := \{\cup \mathcal{C} : \mathcal{C} \subset \mathcal{B}\}$. If a class of subsets of a set generates a topology on the set, it is a base of a topology on the set. The topology generated by a base is the weakest topology containing the base: $\mathcal{T(B)} = \cap_{\mathcal{B} \subset \mathcal{T}} \mathcal{T}$. A topology can have many bases, the largest of which is itself.

Topological space $(X, \mathcal{T})$ is a set $X$ with a topology $\mathcal{T}$. Topological space can be compact or non-compact, connected or disconnected. A set $A$ in a topological space $(X, \mathcal{T})$ is open iff it is an element of the topology: $A \in \mathcal{T}$. Neighborhood $U(x)$ of a point $x$ is an open set containing the point: $x \in U(x) \in \mathcal{T}$.

### Connectedness

Connected set. Disconnected set.

### Compactness

compact

A topological space is locally compact if every point of the space has a compact neighborhood.

A topological space is $\sigma$-compact if it is the union of countably many compact subspaces.

Hausdorff space $T_2$ is a topological space where distinct points have disjoint neighborhoods. (It implies the uniqueness of limits of sequences.) A topology $\mathcal{T}$ of a set $X$ is called Hausdorff if the topological space $(X, \mathcal{T})$ is a Hausdorff space. Metric spaces are Hausdorff. Almost all spaces encountered in analysis are Hausdorff.

### Convergence

Limit point of a set in a topological space is a point in the space each neighbourhood of which intersects with the set at some other point. Derived set $A'$ of a set $A$ in a topological space is the set of all its limit points.

Closed set in a topological space is a set $A$ that contains its derived set: $A' \subset A$. A set is closed iff its complement is open: $\complement A \in \mathcal{T}$. Closure $\bar{A}$ of a set $A$ in a topological space is the union of the set and its derived set: $\bar{A} = A \cup A'$.

Dense set $A$ of a topological space $(X, \mathcal{T})$ is a set whose closure is the entire space: $\bar{A} = X$.

Separable space is a topological space containing a countable dense set. For example, Euclidean spaces are separable.

### Continuous Mapping

Continuous mapping is a mapping $f: X \to Y$ between two topological spaces $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ such that the preimage of any open set is open: $\forall G \in \mathcal{T}_Y, f^{-1}(G) \in \mathcal{T}_X$.

Homeomorphism (同胚) or topological isomorphism (isomorphisms in the category of topological spaces), is a continuous function between two topological spaces that has a continuous inverse function.

Topological embedding is an injective continuous map $f:X \to Y$ between two topological spaces that yields a homeomorphism between $X$ and $f(X)$. An embedding is a representation $f(X)$ of a topological object $X$ in another topological space $Y$, which preserves its connectivity or algebraic properties.

## Manifold

Manifold $M$ is a topological space that is locally Euclidean: each (interior) point has a neighborhood that is homeomorphic to an open ball of a certain dimension; n-manifold $M^n$ is a manifold of dimension $n$. The concept of manifold focuses on "global" properties nonexistent in Euclidean spaces. Manifolds with the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis.

Boundary $\partial M$ of a manifold $M$ is the complement of the interior $\text{Int} M$ of the manifold: $\partial M = M \setminus \text{Int} M$. A boundary point lands on the boundary hyperplane of its neighborhood. Manifold commonly means a compact manifold with boundary, e.g. a sheet of paper is a 2-manifold with a 1-dimensional boundary. Objects that are not manifolds: 8-shaped curve; balloon-shaped surface attached with a line segment.

Submanifold of a manifold is a subset that itself is a manifold of a lower dimension, e.g. closed ball ⊃ sphere ⊃ circle. Whitney embedding theorem: Any manifold can be embedded as a submanifold of a Euclidean space. (But may not be of $n+1$ dimensions, e.g. Klein bottle is a 2-manifold that always self-intersects in 3-dimensional Euclidean space.)

Reach $r$ of a submanifold $M$ of a Euclidean space is the largest real number such that any point $x$ that is a distance less than $r$ from $M$ has a unique projection on $M$.

### Examples

has boundary $\mathbb{R}^2$ $\mathbb{R}^3$ $\mathbb{R}^4$
1-manifold circle trefoil knot
2-manifold sphere, torus; cylinder, Möbius strip Klein bottle

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: $T^2 = S^1 × S^1$.) Applications: symplectic manifold for analytical mechanics (Lagrangian, Hamiltonian) [@Arnold1989]; Lorentzian 4-manifold for general relativity; complex manifold for complex analysis.

### Description and Construction

Coordinate chart $\phi: U \to \mathbb{R}^n$ a homeomorphism from a small neighborhood on a manifold $U \subset M$ to an open subset of a Euclidean space $\phi(U) \in T(\mathbb{R}^n)$, For example, angular coordinate is a chart of a circle, but not a global homeomorphism; in the same sense, geographical coordinates is a chart of a sphere. Transition function $\phi_2 \circ \phi_1^{-1}: \phi_1(U_1 \cap U_2) \to \mathbb{R}^n$ is a map from one coordinate chart $\phi_1$ to another $\phi_2$ on the region they overlap. Atlas $\{\phi\}$ is a class of coordinate charts on a manifold, such that the transition functions of the charts are smooth.

Manifold can be constructed in different ways, depending on the viewpoint:

1. Charts: The manifold is identified as a subset of an embedding Euclidean space, and then an atlas of charts covering this subset is constructed;
2. Surgery theory ("patchwork"): specifying an atlas which is itself defined by transition maps;
3. Quotient space of a manifold that is also a manifold: gluing points together into a single point, such as along boundaries; square to cylinder or Möbius strip, and cylinder to torus or Klein bottle;
4. Product topology: Cartesian products of manifolds, e.g. 1-sphere × 1-sphere is a torus, 1-sphere × line segment is a finite cylinder;

### Invariants

Invariant properties:

• Geometric (local): dimension; curvature (Gauss' theorema egregium);
• Topological (global):
• orientability: true, sphere; false, Möbius strip, Klein bottle, real projective plane;
• simply connectedness (true, sphere; false, torus);
• Euler characteristic, or genus (亏格) for 2-manifolds: 0, sphere; 1, torus;
• Betti numbers, homology (同调), cohomology (上同调).

Simplicial homology (by H. Poincaré); singular homology (by O. Veblen); spectral homology (P.S. Aleksandrov). A singular homology group is an Abelian group which partially counts the number of holes in a topological space.

Classification of manifolds by invariants: It is in general undecidable whether two topological spaces of dimension greater than four are homeomorphic [@Markov1958]. No program can decide whether two 4-manifolds, or of a higher dimension, are diffeomorphic. 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].

### Extensions

A manifold may be endowed with more structure than a locally Euclidean topology. The structure is first defined on each chart separately; if all the transition maps are compatible with this structure, the structure transfers to the manifold.

Smooth manifold is a manifold with a smooth atlas: the transition functions are infinitely differentiable maps from a Euclidean space to itself. Smooth manifold is also known as differentiable manifold, as it allows tangent spaces and calculus on the manifold, see more at Differential Geometry. Diffeomorphism (微分同胚) is a map between smooth manifolds, which is differentiable and has a differentiable inverse.

Riemannian manifold, or Riemannian space, is a smooth manifold with a Riemannian metric (tensor): inner products on tangent spaces that varies smoothly from point to point. Riemannian manifold allows distances and angles on the manifold. Laplace-Beltrami operator $Δ$.

Symplectic manifold is a smooth manifold with a symplectic structure.

### Misc

Scalar-valued functions on manifold. Harmonic analysis of functions, e.g. spherical harmonics.

Directional statistics deals with observations on unit spheres $\mathbb{S}^{d-1}$ [@Brigant2019].

Sampling on manifolds [@Soize2016].

## Algebraic Topology

Algebraic topology studies topological spaces with tools from Abstract Algebra.

### Concepts

Two continuous functions from one topological space to another are homotopic (同伦) if one can be continuously deformed into the other. Such a deformation is a homotopy between the two functions.

Homogeneous space is a set $X$ with a transitive group action $G$: 1. $\forall x, y \in M, \exists g \in G: g x = y$ (transitivity); 1. $e x = x$ (identity map); 1. $(g h) x = g(h x)$ (composition); The elements of $G$ are called the symmetries of $X$.

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 (discrete math) as a topological space is equivalent to simplicial 1-complex.

topological ring (algebra)