nLab
Kolmogorov topological space

Definition

A topological space $(X,\tau)$ is called a Kolmogorov space if it satisfies the $T_0$-separation axiom, hence if for $x_1 \neq x_2 \in X$ any two distinct points, then at least one of them has an open neighbourhood $U_{x_i} \in \tau$ which does not contain the other point.

the main separation axioms

number	name	statement	reformulation
$T_0$	Kolmogorov	given two distinct points, at least one of them has an open neighbourhood not containing the other point	every irreducible closed subset is the closure of at most one point
$T_1$		given two distinct points, both have an open neighbourhood not containing the other point	all points are closed
$T_2$	Hausdorff	given two distinct points, they have disjoint open neighbourhoods	the diagonal is a closed map
$T_{\gt 2}$		$T_1$ and…	all points are closed and…
$T_3$	regular 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
$T_4$	normal 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

Reflection

Proposition

(Kolmogorov quotient)

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

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

Related concepts

• Hausdorff topological space

• nice topological space

References

• Wikipedia, Kolmogorov space