View source for Semiregular space

A '''semiregular space''' is a [[Topological space|topological space]] whose [[Regular open set|regular open set]]s (sets that equal the interiors of their closures) form a [[Base (topology)|base]] for the topology.<ref name="willard"/>
==Examples and sufficient conditions==

Every [[Regular space|regular space]] is semiregular, and every topological space may be embedded into a semiregular space.<ref name="willard">{{citation|title=General Topology|first=Stephen|last=Willard|publisher=Dover|year=2004|isbn=978-0-486-43479-7|page=98|url=https://books.google.com/books?id=-o8xJQ7Ag2cC&pg=PA98|contribution=14E. Semiregular spaces}}.</ref>

The space <math>X = \Reals^2 \cup \{0^*\}</math> with the [[Double origin topology|double origin topology]]<ref>Steen & Seebach, example #74</ref> and the [[Arens square]]<ref>Steen & Seebach, example #80</ref> are examples of spaces that are [[Hausdorff space|Hausdorff]] semiregular, but not regular.

==See also==

* {{annotated link|Separation axiom}}

==Notes==

{{reflist}}

==References==

* 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 General Topology}} <!--{{sfn|Willard|2004|p=}}-->

[[Category:Properties of topological spaces]]
[[Category:Separation axioms]]

{{Sourceattribution|Semiregular space}}