 Regular topological space

A topological space is regular when a point and a closed set not containing it have disjoint open neighborhoods. Equivalently, for every point $x$ in an open set $U$, there is an open $W$ with $x\in W\subseteq \overline{W}\subseteq U$. To obtain shrinking, separate $x$ from the closed complement of $U$; the closure of the point's neighborhood avoids the other neighborhood. Conversely use such a shrinking and the complement of its closure. This definition alone does not require $T_1$.