Regular Open and Closed Sets

Subject: Topology
Links: Interior Points, Limit Points and Closure

We have another classification of sets, that are called regular open and regular closed. A subset $A$ of a topological space $(X,\tau )$ is a regular open set if $\text{int}(\text{cl}(A))=A$. . A subset $F$ of a topological space $(X,\tau )$ is a regular open set if $\text{cl}(\text{int}(F))=F$.

It is equivalent to be a regular open set $A$, and $\text{bd}(\text{cl}(A))=\text{bd}(A)$

It is equivalent to be a regular closed set $A$, and $\text{bd}(\text{int}(A))=\text{bd}(A)$

A subset of $X$ is a regular open set iff its complement in $X$ is a regular closed set.

For any topological space $(X,\tau )$, and any subset $A$ of $X$, the set $\text{int}(\text{cl}(A))$ is a regular open set, and the set $\text{cl}(\text{int}(A))$ is a regular closed set

The intersection of two regular open sets is also a regular open set. Similarly, the union of two regular closed sets is also a regular open set.