nLab accessible topological space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

(0,1)(0,1)-Category theory

Contents

Definition

A topological space (X,τ)(X,\tau) is called an accessible topological space or a T 1T_1-topological space if it is both a Kolmogorov topological space and a symmetric topological space. Or equivalently, if its specialization preorder is equality. Or equivalently still, if all singletons are closed subsets (“points are closed”).

the main separation axioms

numbernamestatementreformulation
T 0T_0Kolmogorovgiven two distinct points, at least one of them has an open neighbourhood not containing the other pointevery irreducible closed subset is the closure of at most one point
T 1T_1given two distinct points, both have an open neighbourhood not containing the other pointall points are closed
T 2T_2Hausdorffgiven two distinct points, they have disjoint open neighbourhoodsthe diagonal is a closed map
T >2T_{\gt 2}T 1T_1 and…all points are closed and…
T 3T_3regular Hausdorff…given a point and a closed subset not containing it, they have disjoint open neighbourhoods…every neighbourhood of a point contains the closure of an open neighbourhood
T 4T_4normal Hausdorff…given two disjoint closed subsets, they have disjoint open neighbourhoods…every neighbourhood of a closed set also contains the closure of an open neighbourhood
… every pair of disjoint closed subsets is separated by an Urysohn function

Properties

In terms of lifting properties

The separation conditions T 0T_0 to T 4T_4 may equivalently be understood as lifting properties against certain maps of finite topological spaces, among others.

This is discussed at separation axioms in terms of lifting properties, to which we refer for further details. Here we just briefly indicate the corresponding lifting diagrams.

In the following diagrams, the relevant finite topological spaces are indicated explicitly by illustration of their underlying point set and their open subsets:

  • points (elements) are denoted by \bullet with subscripts indicating where the points map to;

  • boxes are put around open subsets,

  • an arrow u c\bullet_u \to \bullet_c means that c\bullet_c is in the topological closure of u\bullet_u.

In the lifting diagrams for T 2T 4T_2-T_4 below, an arrow out of the given topological space XX is a map that determines (classifies) a decomposition of XX into a union of subsets with properties indicated by the picture of the finite space.

Notice that the diagrams for T 2T_2-T 4T_4 below do not in themselves imply T 1 T_1 .

Proposition

(Lifting property encoding T 0T_0)
The following lifting property in Top equivalently encodes the separation axiom T 0 T_0 :

Proposition

(Lifting property encoding T 1T_1)
The following lifting property in Top equivalently encodes the separation axiom T 1 T_1 :

Proposition

(Lifting property encoding T 2T_2)
The following lifting property in Top equivalently encodes the separation axiom T 2 T_2 :

Proposition

(Lifting property encoding T 3T_3)
The following lifting property in Top equivalently encodes the separation axiom T 3 T_3 :

Proposition

(Lifting property encoding T 4T_4)
The following lifting property in Top equivalently encodes the separation axiom T 4 T_4 :

Examples

Leibniz's identity of indiscernibles implies that every set XX with its power set 𝒫(X)\mathcal{P}(X) is an accessible topological space:

(P𝒫(X).(xP)(yP))x=y\left(\forall P \in \mathcal{P}(X).(x \in P) \iff (y \in P)\right) \implies x = y

for all xXx \in X and yXy \in X, since the relation

P𝒫(X).(xP)(yP)\forall P \in \mathcal{P}(X).(x \in P) \iff (y \in P)

is an equivalence relation.

preorderpartial orderequivalence relationequality
topological spaceKolmogorov topological spacesymmetric topological spaceaccessible topological space

References

Last revised on January 31, 2024 at 07:43:43. See the history of this page for a list of all contributions to it.