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
$Def: The space X is regular if any xX and CX, with C closed and x not in C, have disjoint neighborhoods. * Result: X is regular if the neighborhood filter of each point has a base consisting of closed sets. Sequential Spaces$ Def: The space (X, T) is sequential if for every open set AX every sequence convergent to a point in A is eventually in A.

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 pX : ∃ {On | n ∈ $$\mathbb N$$, OnT for all n}, such that for all U, pUT : ∃ OnU . * 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.

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 xy, 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.
$Def: A topological space X is normal if for each pair of disjoint closed A, BX there exist disjoint open U, V with AU and BV.$ 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.
* Hausdorff space: A topological space (X, T) such that for every pair of points x, yX there exist neighborhoods U of x and V of y with UV = 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 x0X.
* H-Space: A notion slightly weaker than that of topological group; A pair (X, x0) with a composition law X × XX, 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.