Completely Normal Space

Definition

A Topological Space $(X,\mathcal{T})$ is completely normal ($T_5$) space if it satisfies T1 Axiom and for all Separated Sets $A,B\subset X$, there exists disjoint open sets containing $A$ and $B$ respectively. $\forall A,B\text{s.t.}(\overline{A}\cap B=\varnothing \text{and}A\cap \overline{B}=\varnothing ),\exists U,V\in \mathcal{T}\text{s.t.}U\cap V=\varnothing ,A\subset U,B\subset V$

Facts

A Metric Space is a completely normal space

A topological space $(X,\mathcal{T})$ is a completely normal space if and only if its all subsets $\forall A\subset X$ are normal spaces.