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 ${\displaystyle X=\mathbb {R} ^{2}\cup \{0^{*}\}}$ 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. 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.