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.
Examples and sufficient conditions
Every regular space is semiregular, and every topological space may be embedded into a semiregular space.{{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}}.
The space with the double origin topologySteen & Seebach, example #74 and the Arens squareSteen & Seebach, example #80 are examples of spaces that are Hausdorff semiregular, but not regular.
See also
- {{annotated link|Separation axiom}}
Notes
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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}}