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 $-\infty<a\leq b<\infty$ .
• $(a,b)\cup(b,c)$ is not regular open for $-\infty<a\leq b \leq c<\infty$ 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. $1=X$ and $0=\varnothing$ ,
  2. $a\land b=a\cap b$ ,
  3. $a\lor b=(a\cup b)^{\bot\bot}$ , and
  4. $a^{\prime}=a^{\bot}$ .
  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 $A=\overline{\operatorname{int}(A)}$ .

Bibliography

1

P. Halmos (1970). Lectures on Boolean Algebras, Springer.

2

S. Willard (1970). General Topology, Addison-Wesley Publishing Company.