ThmDex – An index of mathematical definitions, results, and conjectures.
Result R102 on D58: Metric
Singletons are closed in a metric space
Formulation 0
Let $M = (X, \mathcal{T}, d)$ be a D1107: Metric space.
Then \begin{equation} \forall \, x \in X : X \setminus \{ x \} \in X \end{equation}
Proofs
Proof 1
Let $M = (X, \mathcal{T}, d)$ be a D1107: Metric space.
This result is a corollary to the results
(i) R499: Metrisable topological space is Hausdorff
(ii) R517: Singletons are closed in Hausdorff space

$\square$