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

$\square$