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 $\mathbb{ℝ}$ 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 $\mathbb{ℝ}$ 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 $\mathbb{ℝ}$ 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.

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

The interior of a closed subset of $X$ is a regular open subset of $X$ and likewise, the closure of an open subset of $X$ is a regular closed subset of $X.$ 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

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References

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