Let $T = (X, \mathcal{T})$ be a D2299: Polish topological space.

Then
\begin{equation}
\forall \, x \in X :
X \setminus \{ x \} \in \mathcal{T}
\end{equation}

Singletons are closed in Polish space

Formulation 0

Let $T = (X, \mathcal{T})$ be a D2299: Polish topological space.

Then
\begin{equation}
\forall \, x \in X :
X \setminus \{ x \} \in \mathcal{T}
\end{equation}

Proofs

Let $T = (X, \mathcal{T})$ be a D2299: Polish topological space.

This result is a particular case of R102: Singletons are closed in a metric space. $\square$