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}

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}

Subresults

▶ | R4007: Singletons are closed in Polish space |

Proofs

Let $M = (X, \mathcal{T}, d)$ be a D1107: Metric space.

This result is a corollary to the results

$\square$

(i) | R499: Metrisable topological space is Hausdorff |

(ii) | R517: Singletons are closed in Hausdorff space |

$\square$