# regular open set

Let $X$ be a topological space  . A subset $A$ of $X$ is called a regular open set if $A$ is equal to the interior of the closure   of itself:

 $A=\operatorname{int}(\overline{A}).$

Clearly, every regular open set is open, and every clopen set is regular open.

Examples. Let $\mathbb{R}$ be the real line with the usual topology (generated by open intervals  ).

• $(a,b)$ is regular open whenever $-\infty.

• $(a,b)\cup(b,c)$ is not regular open for $-\infty and $a\neq c$. The interior of the closure of $(a,b)\cup(b,c)$ is $(a,c)$.

If we examine the structure  of $\operatorname{int}(\overline{A})$ a little more closely, we see that if we define

 $A^{\bot}:=X-\overline{A},$

then

 $A^{\bot\bot}=\operatorname{int}(\overline{A}).$

So an alternative definition of a regular open set is an open set $A$ such that $A^{\bot\bot}=A$.

Remarks.

The definition of a regular open set can be dualized. A closed set  $A$ in a topological space is called a regular closed set if $A=\overline{\operatorname{int}(A)}$.

## References

• 1 P. Halmos (1970). Lectures on Boolean Algebras, Springer.
• 2 S. Willard (1970). General Topology, Addison-Wesley Publishing Company.
Title regular open set RegularOpenSet 2013-03-22 15:04:03 2013-03-22 15:04:03 CWoo (3771) CWoo (3771) 9 CWoo (3771) Definition msc 06E99 regularly open regularly closed regularly closed set regular open regular closed