Compact Subsets of T3 Spaces

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Theorem

Let $T = \left({S, \tau}\right)$ be a $T_3$ space.

Let $A \subseteq S$ be compact in $T$.


Then for each $U \in \tau$ such that $A \subseteq U$:

$\exists V \in \tau: A \subseteq V \subseteq V^- \subseteq U$

where $V^-$ denotes the closure of $V$.


Proof


Sources