Types of Topological Spaces  

In General > s.a. topology.
* Discrete and indiscrete topology: The discrete topology on a set X is the one in which every subset is open; They can be defined on any set.
* Finite topologies: On 3 points, there are 29; On 4 points, 355; ...; On 14 points, > 1023 connected ones; In general, finite topologies are the same as finite quasi-ordered sets, and every finite topology is generated by a partial pseudometric; Finite T0 spaces are the same as ordered spaces.
@ Finite topologies: Kleitman & Rothschild PAMS(70) [enumeration]; Güldürdek & Richmond Ord(05) [generating pseudometric]; Barmak & Minian math/06 [and finite simplicial complexes]; Ragnarsson & Tenner JCTA(10) [smallest number of points in a topology with k open sets].
@ On discrete sets: Elizalde JMP(87); Hammer in(04)ht/98 [paths, entropy]; > s.a. Geometric Topology.
@ On closed subsets of a compact set: Sorkin & Woolgar CQG(96)gq/95 [Vietoris topology].

Regular Spaces
$ Def: The space X is regular if any x ∈ X and C ⊂ X, with C closed and x not in C, have disjoint neighborhoods.
* Result: X is regular if the neighborhood filter of each point has a base consisting of closed sets.

Sequential Spaces
$ Def: The space (X, T) is sequential if for every open set A ⊂ X every sequence convergent to a point in A is eventually in A.
> Online resources: see Wikipedia page.

First-Countable Spaces
$ Def: The space (X, T) is first countable if each point has a countable base of neighborhoods, i.e., for each point p in X there is a countable collection of open sets such that each neighborhood of p contains at least one of them,

for all p ∈ X : ∃ {On | n ∈ \(\mathbb N\), On ∈ T for all n},  such that  for all U, p ∈ U ∈ T : ∃ On ⊂ U .

* Examples: All metric spaces are first countable (the countable bases of neighborhoods are the balls of radius 1/n).
@ References: Gutierres T&A(06) [without axiom of choice].
> Online resources: see Wikipedia page.

Second-Countable Spaces
$ Def: (X, T) is second countable if it has a countable base.
* Relationships: Every second countable space is first countable, Lindelöf, paracompact.
* Operations: The property is stable under taking a subspace, Cartesian product, countable union.
* Example: \(\mathbb R^n\), with the open balls of rational radius and center, or all rational rectangles, as open sets.
> Online resources: see Wikipedia page.

T-Spaces / Separation Axioms (The terminology comes from "Trennungsaxiom") > s.a. discrete spacetimes; lines [space of causal lines].
* T0 space: Any two distinct points have distinct sets of neighborhoods; Finite ones are in 1-1 correspondence with finite posets.
* T1 space: For any x ≠ y, each has a neighborhood not containing the other; Equivalently, all finite subsets are closed.
* T2 space: See Hausdorff below.
* T3 space: A regular T1 space.
* T4 space: A normal T1 space; Every T4 space is T3; > s.a. Bicompact Space.
* Tychonoff space: A completely regular T1 space.
@ References: Ali-Akbari et al T&A(09) [continuous-poset models of T1 spaces]; Erné T&A(11) [algebraic models for T1 spaces].

Normal Space > s.a. Urysohn Lemma.
$ Def: A topological space X is normal if for each pair of disjoint closed A, B ⊂ X there exist disjoint open U, V with A ⊂ U and B ⊂ V.
$ Alternative def: A space X is normal iff for all neighborhoods U of a closed C, ∃ V neighborhood of C, such that closure(V) ⊂ U.

Other Types > s.a. 2D, 3D and 4D manifolds; compact spaces; connected spaces; posets [topological orders].
* Hausdorff space: A topological space (X, T) such that for every pair of points x, y ∈ X there exist neighborhoods U of x and V of y with U ∩ V = Ø; There is an equivalent characterization in terms of X × X, and one in terms of filters (no filter has more than one limit point); > s.a. Bicompact Space; topology [set of Hausdorff topologies on X]; topology in physics.
* Lindelöf: A topological space has the Lindelöf property if every open cover has a countable subcover; > s.a. spacetime topology [example of non-Lindelöf]; generalized posets.
* Perfect: One every point of which is an accumulation point.
* Separable: One that has a countable dense subset?
* Small loop space: A topological space in which every loop is small (> see loops).
* Stone space: A "Boolean space", a totally disconnected compact Hausdorff space; Dual to Boolean algebras.
* CO space: A topological space X such that every closed subset of X is homeomorphic to some clopen subset of X; For example, every ordinal with its order topology is a CO space.
@ References: Johnstone 86 [Stone space]; Moore T&A(05) [locally compact locally countable, interesting-sounding mumbo-jumbo]; Bonnet & Rubin T&A(08) [CO spaces]; Richmond T&A(08) [principal topologies, and transformation semigroups]; Heller et al JMP(11)-a1007 [physical significance of non-Hausdorff spaces]; de Seguins Pazzis T&A(13) [the geometric realization of a simplicial Hausdorff space is Hausdorff]; Porchon T&IA(13) [expanding topological spaces].
> Other: see Alexandroff Spaces; Contractible Spaces; paracompact spaces; Topological Manifold; Toronto Space; Triangulable Spaces.

Pointed Topological Spaces
* Idea: A pair (X, x0), where X is a topological space and x0 ∈ X.
* H-Space: A notion slightly weaker than that of topological group; A pair (X, x0) with a composition law X × X → X, such that (i) the composition map is continuous; (ii) x0x0 = x0; (iii) ...
* Examples: Topological groups, with x0 = e.

Examples > s.a. spacetime topology; topology [topologies induced by other structures].
* Topologies on Rn: The usual one is induced by the Euclidean norm, or any of the Lp norms, but inequivalent topologies can be induced, e.g, by first mapping \(\mathbb R\)n to \(\mathbb R\)m by some point set bijection (combining coordinates into fewer ones or splitting them to obtain more) and then pulling back the one on \(\mathbb R\)m.
$ Comb space: The set C:= ({1/n | n ∈ \(\mathbb N\)}× I) ∪ (I × {0}) ∪ ({0} × I), where I:= [0,1].
> Other: see distance between metrics [topology on space of metrics]; lorentzian geometries; riemannian geometries.

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