A subset

S

of a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

X

is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if

\operatorname(\overline) = S

or, equivalently, if

\partial(\overline)=\partial S,

where

\operatorname S,

\overline

and

\partial S

denote, respectively, the interior, closure and boundary of

S.

Steen & Seebach, p. 6 A subset

S

X

is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if

\overline = S

or, equivalently, if

\partial(\operatornameS)=\partial S.

Examples

\Reals

has its usual

Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdot ...

then the open set

S = (0,1) \cup (1,2)

is not a regular open set, since

\operatorname(\overline) = (0,2) \neq S.

Every

open interval In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...

\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

\

is a closed subset of

\R

but not a regular closed set because its interior is the empty set

\varnothing,

so that

\overline = \overline = \varnothing \neq \.

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 In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...

and every regular closed set is a

closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...

. 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.

Willard, "3D, Regularly open and regularly closed sets", p. 29 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 Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...

operation is given by

U \vee V = \operatorname(\overline),

the meet is

U \and V = U \cap V

and the complement is

\neg U = \operatorname(X \setminus U).

Notes

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). * General topology

Examples

Properties

See also

Notes

References