regular open set

Let $X$ be a topological space. A subset $A$ of $X$ is called aregular 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 $ℝ$ be the real line with the usualtopology (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 $\mathrm{int}(\overline{A})$ alittle 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 ofa 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 $X^{\perp }=\mathrm{\varnothing }$.

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

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

• It can be shown that if $A$ is open, then $A^{\perp }$ isregular 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.

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

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

  1. $1=X$ and $0=\mathrm{\varnothing }$,

  2. $a\wedge b=a\cap b$,

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

  4. $a^{\prime }=a^{\perp }$.

  This is an example of a Boolean algebra coming from a collection ofsubsets of a set that is not formed by the standard set operationsunion $\cup$, 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 $A=\overline{\mathrm{int}(A)}$.