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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 {\displaystyle \operatorname {Int} =S} or, equivalently, if ∂ = ∂ S , {\displaystyle \partial =\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.}
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 ∂ = ∂ S . {\displaystyle \partial =\partial S.}