Types
of Topological Spaces |

**In General** > s.a. topology.

* __Discrete and indiscrete
topology__: They can be defined on any set.

* __Finite topologies__:
On 3 points, there are 29; On 4 points, 355; ...; On 14 points, > 10^{23} 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 T_{0} 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

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**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* : ∃ {*O*_{n} | *n* ∈ \(\mathbb N\), *O*_{n}
∈ *T* for all *n*}, such
that for all *U*, *p* ∈ *U* ∈ *T* :
∃ *O*_{n} ⊂ *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].

* T_{0} __space__:
Any two distinct points have distinct sets of neighborhoods; Finite ones are
in 1-1 correspondence with finite posets.

* T_{1} __space__: For any *x* ≠ *y*,
each has a neighborhood not containing the other;
Equivalently, all finite subsets are closed.

* T_{2} __space__: See Hausdorff below.

* T_{3} __space__: A regular T_{1} space.

* T_{4} __space__:
A normal T_{1} space; Every T_{4} space
is T_{3}; > s.a.
Bicompact Space.

* __Tychonoff space__: A completely regular T_{1} space.

@ __References__: Ali-Akbari et al T&A(09) [continuous-poset models of T_{1} spaces]; Erné T&A(11) [algebraic models for T_{1} 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.

* __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*,* x*_{0}),
where *X* is a topological space and *x*_{0} ∈ *X*.

* __H-Space__: A notion slightly
weaker than that of topological group; A
pair (*X*,* x*_{0}) with a composition
law *X* × *X* → *X*, such that (i) the composition map is continuous;
(ii) *x*_{0}*x*_{0} = *x*_{0};
(iii) ...

* __Examples__: Topological groups,
with *x*_{0} = *e*.

**Examples** > s.a. spacetime topology;
topology [topologies induced by other structures].

* __Topologies on R__

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