Regular open set explained

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 \operatorname{Int}(\overline{S}) = S or, equivalently, if \partial(\overline{S}) = \partial S, where \operatorname{Int} S, \overline{S} and \partial S denote, respectively, the interior, closure and boundary of S. ^[1]

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{\operatorname{Int} S} = S or, equivalently, if \partial(\operatorname{Int}S) = \partial S. ^[1]

Examples

If

\Reals has its usual Euclidean topology then the open set S = ( 0 , 1 ) \cup ( 1 , 2 ) is not a regular open set, since \operatorname{Int}(\overline{S}) = ( 0 , 2 ) ≠ 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{\operatorname{Int} \{x\}} = \overline{\varnothing} = \varnothing ≠ \{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. ^[2] 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.^[2]

The collection of all regular open sets in

X forms a complete Boolean algebra; the join operation is given by U \vee V = \operatorname{Int}(\overline{U \cup V}), the meet is U ∧ V = U \cap V and the complement is \neg U = \operatorname{Int}(X \setminus U).

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).

Notes and References

1. Steen & Seebach, p. 6
2. Willard, "3D, Regularly open and regularly closed sets", p. 29