ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2748 on D98: Closed set
Finite sets are closed in metric space
Formulation 0
Let $M = (X, \mathcal{T}, d)$ be a D1107: Metric space such that
(i) $E \subseteq X$ is a D78: Subset of $X$
(ii) $E$ is a D17: Finite set
Then \begin{equation} X \setminus E \in \mathcal{T} \end{equation}
Proofs
Proof 0
Let $M = (X, \mathcal{T}, d)$ be a D1107: Metric space such that
(i) $E \subseteq X$ is a D78: Subset of $X$
(ii) $E$ is a D17: Finite set
Since we can partition a finite set into a finite union of singletons, this claim turns out to be a consequence of the results
(i) R102: Singletons are closed in a metric space
(i) R74: Finite union of closed sets is closed

$\square$