Regular Space

Definition

A Topological Space $(X,\mathcal{T})$ is regular ($T_3$) space if it satisfies T1 Axiom and for all disjoint $x$ and Closed Set $B$ in $X$, there exists disjoint open sets containing $x$ and $B$ respectively. $\forall x\in X,(X\setminus B)\in \mathcal{T}\text{s.t.}x\cap B=\varnothing ,\exists U,V\in \mathcal{T}\text{s.t.}U\cap V=\varnothing ,x\in U,B\subset V$

Or equivalently, $\forall x\in U\in \mathcal{T},\exists {\mathcal{N}}_x\text{s.t.}{\bar{\mathcal{N}}}_x\subset U$

Facts

A Subspace Topology of a regular space is a regular space.

The Product Topology of any collection of regular spaces is a regular space.

A regular Lindelof Space is a Normal Space.

A Locally Compact Hausdorff Space is a regular space.