Regular space

Definition

A topological space $X$ is regular when the property that follows holds. For each closed set $F \subseteq X$ and each point $x \in X \setminus F$, there are open sets $U$ and $V$ with $x \in U$, $F \subseteq V$ and $U \cap V = \varnothing$.

Remarks

Names vary. Some authors call a space T3 when it is regular and T0, and then it is also Hausdorff. Other authors include this condition in the word regular. Equivalently, $X$ is regular if and only if each neighbourhood of a point $x$ contains a closed neighbourhood of $x$. Each metric space is regular. Each normal T1 space is regular.

Explanations

L1 human

In a regular space, you can separate a point from a closed set that does not contain it, by two disjoint open sets. Hausdorff spaces only separate points from points.