nLab
Kolmogorov 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

Contents

Definition

A topological space (X,τ)(X,\tau) is called a Kolmogorov space if it satisfies the T 0T_0-separation axiom, hence if for x 1x 2Xx_1 \neq x_2 \in X any two distinct points, then at least one of them has an open neighbourhood U x iτU_{x_i} \in \tau which does not contain the other point.

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 :

Reflection

Proposition

(Kolmogorov quotient)

Let (X,τ)(X,\tau) be a topological space. Consider the relation on the underlying set by which x 1x 1x_1 \sim x_1 precisely if neighther x ix_i has an open neighbourhood not containing the other. This is an equivalence relation. The quotient topological space XX/X \to X/\sim by this equivalence relation is a T 0T_0-space.

This construction is the reflector exhibiting Kolmogorov spaces as a reflective subcategory of the category Top of all topological spaces.

References

Last revised on October 19, 2021 at 06:44:10. See the history of this page for a list of all contributions to it.