Regular open set

A subset <math>S</math> of a topological space <math>X</math> is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if <math>\operatorname{Int}(\overline{S}) = S</math> or, equivalently, if <math>\partial(\overline{S})=\partial S,</math> where <math>\operatorname{Int} S,</math> <math>\overline{S}</math> and <math>\partial S</math> denote, respectively, the interior, closure and boundary of <math>S.</math><ref name="S&Sp6">Steen & Seebach, p. 6</ref>

A subset <math>S</math> of <math>X</math> is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if <math>\overline{\operatorname{Int} S} = S</math> or, equivalently, if <math>\partial(\operatorname{Int}S)=\partial S.</math><ref name="S&Sp6"/>

Examples

If <math>\Reals</math> has its usual Euclidean topology then the open set <math>S = (0,1) \cup (1,2)</math> is not a regular open set, since <math>\operatorname{Int}(\overline{S}) = (0,2) \neq S.</math> Every open interval in <math>\R</math> 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 <math>\{x\}</math> is a closed subset of <math>\R</math> but not a regular closed set because its interior is the empty set <math>\varnothing,</math> so that <math>\overline{\operatorname{Int} \{x\}} = \overline{\varnothing} = \varnothing \neq \{x\}.</math>

Properties

A subset of <math>X</math> is a regular open set if and only if its complement in <math>X</math> is a regular closed set.<ref name="willard-regopen"/> Every regular open set is an open set and every regular closed set is a closed set.

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

The interior of a closed subset of <math>X</math> is a regular open subset of <math>X</math> and likewise, the closure of an open subset of <math>X</math> is a regular closed subset of <math>X.</math><ref name="willard-regopen">Willard, "3D, Regularly open and regularly closed sets", p. 29</ref> 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.<ref name="willard-regopen"/>

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

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