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

Result R2746 on D98: Closed set

Finite sets are closed in Fréchet space

Formulation 0

Let $T = (X, \mathcal{T})$ be a D784: Fréchet topological space.
Let $E \subseteq X$ be 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 D784: Fréchet topological space.
Let $E \subseteq X$ be a D17: Finite set.

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

(i)	R457: Fréchet space iff singletons are closed
(ii)	R74: Finite union of closed sets is closed

$\square$