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Regular open set

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A subset of a topological space is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if or, equivalently, if where and denote, respectively, the interior, closure and boundary of [1]

A subset of is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if or, equivalently, if [1]

Examples

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If has its usual Euclidean topology then the open set is not a regular open set, since Every open interval in 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 but not a regular closed set because its interior is the empty set so that

Properties

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A subset of is a regular open set if and only if its complement in is a regular closed set.[2] Every regular open set is an open set and every regular closed set is a closed set.

Each clopen subset of (which includes and itself) is simultaneously a regular open subset and regular closed subset.

The interior of a closed subset of is a regular open subset of and likewise, the closure of an open subset of is a regular closed subset of [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 forms a complete Boolean algebra; the join operation is given by the meet is and the complement is

See also

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  • List of topologies – List of concrete topologies and topological spaces
  • Regular space – topological space in which a point and a closed set are, if disjoint, separable by neighborhoods
  • Semiregular space
  • Separation axiom – Axioms in topology defining notions of "separation"

Notes

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

References

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  • 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.