Semiregular space

A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base.[1] Equivalently, it is any topological space for which the set of all regular open subsets forms a base.

Examples and sufficient conditions

Every regular space is semiregular, and every topological space may be embedded into a semiregular space.[1]

The space with the double origin topology[2] and the Arens square[3] are examples of spaces that are Hausdorff semiregular, but not regular.

Notes
  1. Willard, Stephen (2004), "14E. Semiregular spaces", General Topology, Dover, p. 98, ISBN 978-0-486-43479-7.
  2. Steen & Seebach, example #74
  3. Steen & Seebach, example #80

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, Stephen (2004) [1970]. General Topology (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
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