T₀ space - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A $T_0$ space is a topological space $X$ such that for any distinct points $x,y\in X$ , there is an open set containing exactly one of them.

Comparison with stronger axioms

This is the weakest of the common separation axioms. Every $T_1$ space, and hence every Hausdorff space, is $T_0$ .

Examples

The Sierpiński space on $\{0,1\}$ , with open sets $\varnothing$ , $\{1\}$ , and $\{0,1\}$ , is $T_0$ but not $T_1$ .
Every metric space is $T_0$ , since it is Hausdorff.