Polished A-A and added new lines for broken enumerates.
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Bokuan Li
2026-05-05 01:50:35 -04:00
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\begin{enumerate}
\item For each $x \in X$ and $A \subset X$ closed with $x \not\in A$, there exists $U \in \cn(x)$ and $V \in \cn(A)$ such that $U \cap V = \emptyset$.
\item For each $x \in X$, the closed neighbourhoods of $x$ forms a fundamental system of neighbourhoods at $x$.
\end{enumerate}
\end\{enumerate\}
If $X$ is a T1 space such that the above holds, then $X$ is \textbf{regular}.
\end{definition}
\begin{proof}[Proof {{\cite[Proposition 1.4.11]{Bourbaki}}}. ]