PlanetMath: regular open set

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regular open set

(Definition)

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

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

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).

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 .

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

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

then

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

So an alternative definition of a regular open set is an open set

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 .
$A^{\bot}\cup B^{\bot}\subseteq(A\cap B)^{\bot}$ and $A^{\bot}\cap B^{\bot}=(A\cup B)^{\bot}$ .
It can be shown that if 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 of .
In addition, if both and 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 forms a Boolean algebra under the following set of operations:
1. 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 in a topological space is called a regular closed set if $A=\overline{\operatorname{int}(A)}$ .