Types of Topological Spaces

In General [> s.a. topology.]
* Discrete and indiscrete topology: 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; T₀ spaces are the same as ordered spaces.
* On Rⁿ: The Euclidean topology is induced by any of the L^p norms.
@ Finite topologies: Kleitman & Rothschild PAMS(70) [enumeration]; 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 compact set: Sorkin & Woolgar CQG(96)gq/95 [Vietoris topology].

Regular Spaces
$ Def: The space X is regular if any x and C, 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.

First-Countable Spaces
$ Def: The space (X, T) is first countable if 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 (each point has a countable base of neighborhoods),

for all p X : {O_n | n N, O_n T for all n},  such that  for all U, p U T : O_n U .

@ References: Gutierres T&A(06) [without axiom of choice].

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: Rⁿ, with open sets the open balls of rational radius and center, or all rational rectangles.

T-Spaces (The terminology comes from "Trennungsaxiom") > s.a. discrete spacetimes.
* T₀ space: Any two distinct points have distinct sets of neighborhoods.
* 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].

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.
* 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].
* 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]; 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].
> Other: see compact spaces; connected spaces; Contractible Spaces; paracompact spaces; Topological Manifold; 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 [induced by other structures].
* Topologies on Rⁿ: The usual one is induced by the Euclidean norm, but inequivalent topologies can be induced, e.g, by first mapping Rⁿ to 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 R^m.
$ Comb space: The set C:= ({1/n | n N}× I) (I × {0}) ({0} × I), where I:= [0,1].

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