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]

Examples and sufficient conditions

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

The space X = R 2 ∪ { 0 ∗ } {\displaystyle X=\mathbb {R} ^{2}\cup \{0^{*}\}} with the double origin topology[2] and the Arens square[3] are examples of spaces that are Hausdorff semiregular, but not regular.

See also

  • Separation axiom – Axioms in topology defining notions of "separation"

Notes

  1. ^ a b 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. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.