ThmDex – An index of mathematical definitions, results, and conjectures.
Singletons are closed in Hausdorff space
Formulation 0
Let $T = (X, \mathcal{T})$ be a D465: Hausdorff topological space.
Then \begin{equation} \forall \, x \in X : X \setminus \{ x \} \in \mathcal{T} \end{equation}
Proofs
Proof 0
Let $T = (X, \mathcal{T})$ be a D465: Hausdorff topological space.
This result is a corollary to the results
(1) R519: Hausdorff space is Fréchet space
(2) R457: Fréchet space iff singletons are closed

$\square$