# 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.

• For any $A\subseteq X$, $A^{\bot}$ is always open.

• $\varnothing^{\bot}=X$ and $X^{\bot}=\varnothing$.

• $A\cap A^{\bot}=\varnothing$ and $A\cup A^{\bot}$ is dense in $X$.

• $A^{\bot}\cup B^{\bot}\subseteq(A\cap B)^{\bot}$ and $A^{\bot}\cap B^{\bot}=(A\cup B)^{\bot}$.

• It can be shown that if $A$ is open, then $A^{\bot}$ is regular open. As a result, following from the first property, $\operatorname{int}(\overline{A})$, being $A^{\bot\bot}$, is regular open for any subset $A$ of $X$.

• In addition, if both $A$ and $B$ are regular open, then $A\cap B$ is regular open.

• It is not true, however, that the union of two regular open sets is regular open, as illustrated by the second example above.

• It can also be shown that the set of all regular open sets of a topological space $X$ forms a Boolean algebra under the following set of operations:

1. (a)

$1=X$ and $0=\varnothing$,

2. (b)

$a\land b=a\cap b$,

3. (c)

$a\lor b=(a\cup b)^{\bot\bot}$, and

4. (d)

$a^{\prime}=a^{\bot}$.

This is an example of a Boolean algebra coming from a collection of subsets of a set that is not formed by the standard set operations union $\cup$, intersection $\cap$, and complementation ${}^{\prime}$.

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