Semiregular space

A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base for the topology.[1]

## Contents

* 1 Properties and examples
* 2 See also
* 3 Notes
* 4 References

## Properties and examples

Every regular space is semiregular.[1] The converse is not true. For example, the space X=ℝ2∪{0*} with the double origin topology[2] and the Arens square[3] are Hausdorff semiregular spaces that are not regular.

Open subspaces of a semiregular space are semiregular.[4] But arbitrary subspaces, even closed subspaces, need not be semiregular.[4]

The product of an arbitrary family of semiregular spaces is semiregular.[4]

Every topological space may be embedded into a semiregular space.[1]

## See also

* Separation axiom - Axioms in topology defining notions of "separation"

## Notes

1. ↑ 1.0 1.1 1.2 Willard, Stephen (2004), "14E. Semiregular spaces", General Topology, Dover, p. 98, ISBN 978-0-486-43479-7, https://books.google.com/books?id=-o8xJQ7Ag2cC&pg=PA98 .
2. ↑ Steen & Seebach, example #74
3. ↑ Steen & Seebach, example #80
4. ↑ 4.0 4.1 4.2 Engelking 1989, Problem 2.7.6(b).

## References

* Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4.
* 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). General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.

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