T1 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 T1 space is a topological space $X$ such that for every point $x\in X$ , the singleton $\{x\}$ is a closed set in $X$ .

Equivalent characterizations

Equivalently, for any distinct points $x\neq y$ there exists an open set containing $x$ but not $y$ , and (symmetrically) an open set containing $y$ but not $x$ .

Remarks

The T1 axiom strengthens T0 and is implied by the Hausdorff condition.

Examples

Every metric space is T1.
On an infinite set $X$ , the cofinite topology (open sets are $\varnothing$ and complements of finite sets) is T1 but not Hausdorff.