Topology is the invariant property of objects under continuous transformations (stretch, twist).
Embedding is a representation of a topological object in a certain space, which preserves its connectivity or algebraic properties.
Point-set topology is the study of the general abstract nature of continuity on spaces. Concepts: continuity, dimension, compactness, and connectedness.
Other types of topology: graph (discrete math), field (algebra).
Topological space \((X, T)\) is a set with a collection of (open/closed) subsets, such that: the empty set and the full set are elements of the collection; and the collection is closed in (finite/arbitrary) intersection and (arbitrary/finite) union.
Topological space can be compact or non-compact, connected or disconnected.
Neighborhood of a point is an open set containing the point.
Closure.
Connected set; disconnected set.
Dense set; Separable set;
A topology is said to be weaker than another if the former specifies a coarser structure of the underlying set. Symbolically, \( \tau_1 \) is weaker than \( \tau_2 \) if \( \tau_1 \subset \tau_2 \). The weakest topology is the collection of the empty set and the full set.
Limit
Inverse image of an open set is open.
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.
A Hausdorff space is a topological space where distinct points have disjoint neighborhoods. It implies the uniqueness of limits of sequences. Almost all spaces encountered in analysis are Hausdorff; more generally, all metric spaces are Hausdorff.
Manifold 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. The concept of manifold focuses on "global" properties nonexistent in Euclidean spaces. N-manifold is a manifold of dimension n. The boundary of a manifold is the complement of the interior of the manifold: \( \partial M = M \setminus \text{Int} M \). 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. A submanifold is a subset of a manifold that is itself a manifold of a smaller dimension, e.g. closed ball ⊃ sphere ⊃ circle. Whitney embedding theorem: Any manifold can be embedded as a submanifold of an 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.)
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 could be much more complex than an 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 torus.) Applications: symplectic manifold for analytical mechanics (Lagrangian, Hamiltonian); Lorentzian 4-manifold for general relativity; complex manifold for complex analysis.
Coordinate chart is a local map of points to coordinates: a homeomorphism from a small neighborhood of a manifold to an open subset of an Euclidean space, \( \phi: U \to \mathbb{R}^n \) where \( U \subset M \) and \(\phi(U) \in T(\mathbb{R}^n) \). 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 is a map from one coordinate chart to another on the region they overlap: \( \phi_2 \circ \phi_1^{-1}: \phi_1(U_1 \cap U_2) \to \mathbb{R}^n \). Atlas is a collection of coordinate charts on a manifold, such that the transition functions of the charts are smooth: \( {\phi} \).
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 an Euclidean space to itself. Smooth manifold allows tangent spaces and calculus on the manifold, also known as differentiable manifold. Riemannian manifold is a smooth manifold with a Riemannian metric: inner products on tangent spaces that varies smoothly from point to point. Riemannian manifold allows distances and angles on the manifold. Symplectic manifold is a smooth manifold with a symplectic structure.
Manifold can be constructed in different ways, depending on the viewpoint:
Invariants:
Classification of manifolds by invariants: No program can decide whether two 4-manifolds, or of a higher dimension, are diffeomorphic.
Scalar-valued functions on manifold: spherical harmonics.
Directional statistics deals with observations on compact Riemannian manifolds: sampling on manifolds {Soize2016}.
Manifolds with the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis.