ThmDex – An index of mathematical definitions, results, and conjectures.

Result R2747 on D98: Closed set

Finite sets are closed in Hausdorff space

Formulation 0

Let $T = (X, \mathcal{T})$ be a D465: Hausdorff topological space such that

(i)	$E \subseteq X$ is a D17: Finite set

Then

(1)	$E$ is a D98: Closed set in $T$
(2)	$X \setminus E$ is a D97: Open set in $T$

Proofs

Proof 0

Let $T = (X, \mathcal{T})$ be a D465: Hausdorff topological space such that

(i)	$E \subseteq X$ is a D17: Finite set

The second claim follows from the first one by definition. The first claim is a corollary to the results

(i)	R517: Singletons are closed in Hausdorff space
(i)	R74: Finite union of closed sets is closed

$\square$