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:
$$A=\mathrm{int}(\overline{A}).$$ 
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 $$.

•
$(a,b)\cup (b,c)$ is not regular open for $$ and $a\ne c$. The interior of the closure of $(a,b)\cup (b,c)$ is $(a,c)$.
If we examine the structure^{} of $\mathrm{int}(\overline{A})$ a little more closely, we see that if we define
$${A}^{\perp}:=X\overline{A},$$ 
then
$${A}^{\perp \perp}=\mathrm{int}(\overline{A}).$$ 
So an alternative definition of a regular open set is an open set $A$ such that ${A}^{\perp \perp}=A$.
Remarks.

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

•
${\mathrm{\varnothing}}^{\perp}=X$ and ${X}^{\perp}=\mathrm{\varnothing}$.

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

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

•
It can be shown that if $A$ is open, then ${A}^{\perp}$ is regular open. As a result, following from the first property, $\mathrm{int}(\overline{A})$, being ${A}^{\perp \perp}$, 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.

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

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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^{}:

(a)
$1=X$ and $0=\mathrm{\varnothing}$,

(b)
$a\wedge b=a\cap b$,

(c)
$a\vee b={(a\cup b)}^{\perp \perp}$, and

(d)
${a}^{\prime}={a}^{\perp}$.
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 ${}^{\prime}$.

(a)
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 $A=\overline{\mathrm{int}(A)}$.
References
 1 P. Halmos (1970). Lectures on Boolean Algebras, Springer.
 2 S. Willard (1970). General Topology, AddisonWesley Publishing Company.
Title  regular open set 

Canonical name  RegularOpenSet 
Date of creation  20130322 15:04:03 
Last modified on  20130322 15:04:03 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06E99 
Synonym  regularly open 
Synonym  regularly closed 
Synonym  regularly closed set 
Defines  regular open 
Defines  regular closed 