Category:T3 Spaces

From ProofWiki
Jump to navigation Jump to search

This category contains results about $T_3$ spaces in the context of Topology.


$T = \left({S, \tau}\right)$ is a $T_3$ space if and only if:

$\forall F \subseteq S: \complement_S \left({F}\right) \in \tau, y \in \complement_S \left({F}\right): \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \varnothing$

That is, for any closed set $F \subseteq S$ and any point $y \in S$ such that $y \notin F$ there exist disjoint open sets $U, V \in \tau$ such that $F \subseteq U$, $y \in V$.