Regular open set

A subset S {\displaystyle S} of a topological space X {\displaystyle X} is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if Int ⁡ ( S ¯ ) = S {\displaystyle \operatorname {Int} ({\overline {S}})=S} or, equivalently, if ∂ ( S ¯ ) = ∂ S , {\displaystyle \partial ({\overline {S}})=\partial S,} where Int ⁡ S , {\displaystyle \operatorname {Int} S,} S ¯ {\displaystyle {\overline {S}}} and ∂ S {\displaystyle \partial S} denote, respectively, the interior, closure and boundary of S . {\displaystyle S.} ^[1]

A subset S {\displaystyle S} of X {\displaystyle X} is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if Int ⁡ S ¯ = S {\displaystyle {\overline {\operatorname {Int} S}}=S} or, equivalently, if ∂ ( Int ⁡ S ) = ∂ S . {\displaystyle \partial (\operatorname {Int} S)=\partial S.} ^[1]

If R {\displaystyle \mathbb {R} } has its usual Euclidean topology then the open set S = ( 0 , 1 ) ∪ ( 1 , 2 ) {\displaystyle S=(0,1)\cup (1,2)} is not a regular open set, since Int ⁡ ( S ¯ ) = ( 0 , 2 ) ≠ S . {\displaystyle \operatorname {Int} ({\overline {S}})=(0,2)\neq S.} Every open interval in R {\displaystyle \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 } {\displaystyle \{x\}} is a closed subset of R {\displaystyle \mathbb {R} } but not a regular closed set because its interior is the empty set ∅ , {\displaystyle \varnothing ,} so that Int ⁡ { x } ¯ = ∅ ¯ = ∅ ≠ { x } . {\displaystyle {\overline {\operatorname {Int} \{x\}}}={\overline {\varnothing }}=\varnothing \neq \{x\}.}

A subset of X {\displaystyle X} is a regular open set if and only if its complement in X {\displaystyle X} is a regular closed set.^[2] Every regular open set is an open set and every regular closed set is a closed set.

A subset G {\displaystyle G} in a topological space X {\displaystyle X} is a regular open set if and only if G = Int ⁡ ( A ¯ ) {\displaystyle G=\operatorname {Int} ({\overline {A}})} for some A ⊂ X {\displaystyle A\subset X} ^[2]. This is a consequence of the maximal and minimal properties of the interior and closure operators which when combined, they lead to

Int ⁡ ( A ¯ ) ⊂ Int ⁡ ( A ¯ ) ¯ ⟹ Int ⁡ ( A ¯ ) ⊂ Int ⁡ ( Int ⁡ ( A ¯ ) ¯ ) {\displaystyle {\begin{aligned}\operatorname {Int} ({\overline {A}})\subset {\overline {\operatorname {Int} ({\overline {A}})}}\quad \Longrightarrow \quad \operatorname {Int} ({\overline {A}})\subset \operatorname {Int} {\Big (}{\overline {\operatorname {Int} ({\overline {A}})}}{\Big )}\end{aligned}}}

Int ⁡ ( A ¯ ) ⊂ A ¯ ⟹ Int ⁡ ( A ¯ ) ¯ ⊂ A ¯ ⟹ Int ⁡ ( Int ⁡ ( A ¯ ) ¯ ) ⊂ Int ⁡ ( A ¯ ) {\displaystyle {\begin{aligned}\operatorname {Int} ({\overline {A}})\subset {\overline {A}}\quad \Longrightarrow \quad {\overline {\operatorname {Int} ({\overline {A}})}}\subset {\overline {A}}\quad \Longrightarrow \quad \operatorname {Int} {\Big (}{\overline {\operatorname {Int} ({\overline {A}})}}{\Big )}\subset \operatorname {Int} ({\overline {A}})\end{aligned}}}

Each clopen subset of X {\displaystyle X} (which includes ∅ {\displaystyle \varnothing } and X {\displaystyle 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.^[2]

The collection of all regular open sets in X {\displaystyle X} forms a complete Boolean algebra; the join operation is given by U ∨ V = Int ⁡ ( U ∪ V ¯ ) , {\displaystyle U\vee V=\operatorname {Int} ({\overline {U\cup V}}),} the meet is U ∧ V = U ∩ V {\displaystyle U\land V=U\cap V} and the complement is ¬ U = Int ⁡ ( X ∖ U ) . {\displaystyle \neg U=\operatorname {Int} (X\setminus U).}

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

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.