Regular open set

A subset $S$ of a topological space $X$ is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if $\mathrm{Int}(\bar{S})=S$ or, equivalently, if $\partial (\bar{S})=\partial S,$ where $\mathrm{Int}S,$ $\bar{S}$ and $\partial S$ denote, respectively, the interior, closure and boundary of $S.$

A subset $S$ of $X$ is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if $\overline{\mathrm{Int}S}=S$ or, equivalently, if $\partial (\mathrm{Int}S)=\partial S.$

Examples

If $ℝ$ has its usual Euclidean topology then the open set $S=(0,1)\cup (1,2)$ is not a regular open set, since $\mathrm{Int}(\bar{S})=(0,2)\ne S.$ Every open interval in $ℝ$ is a regular open set and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed set. A singleton $\{x\}$ is a closed subset of $ℝ$ but not a regular closed set because its interior is the empty set $\varnothing ,$ so that $\overline{\mathrm{Int}\{x\}}=\bar{\varnothing }=\varnothing \ne \{x\}.$

Properties

A subset of $X$ is a regular open set if and only if its complement in $X$ is a regular closed set. Every regular open set is an open set and every regular closed set is a closed set.

A subset $G$ in a topological space $X$ is a regular open set if and only if $G=\mathrm{Int}(\bar{A})$ for some $A\subset X$. This is a consequence of the maximal and minimal properties of the interior and closure operators which when combined, they lead to

$\begin{array}{c}\mathrm{Int}(\bar{A})\subset \overline{\mathrm{Int}(\stackrel{‾}{A})}⟹\mathrm{Int}(\bar{A})\subset \mathrm{Int}\left(\overline{\mathrm{Int}(\stackrel{‾}{A})}\right)\end{array}$

$\begin{array}{c}\mathrm{Int}(\bar{A})\subset \bar{A}⟹\overline{\mathrm{Int}(\stackrel{‾}{A})}\subset \bar{A}⟹\mathrm{Int}\left(\overline{\mathrm{Int}(\stackrel{‾}{A})}\right)\subset \mathrm{Int}(\bar{A})\end{array}$

Each clopen subset of $X$ (which includes $\varnothing$ and $X$ itself) is simultaneously a regular open subset and regular closed subset.

The intersection (but not necessarily the union) of two regular open sets is a regular open set. Similarly, the union (but not necessarily the intersection) of two regular closed sets is a regular closed set.

The collection of all regular open sets in $X$ forms a complete Boolean algebra; the join operation is given by $U\vee V=\mathrm{Int}(\overline{U\cup V}),$ the meet is $U\wedge V=U\cap V$ and the complement is $\mathrm{\neg }U=\mathrm{Int}(X\setminus U).$

See also

• Regular space – Property of topological space
• Separation axiom – Axioms in topology defining notions of "separation"

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).