Semiregular space explained

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.^[1]

The space

X = \Reals ^{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.

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).

Notes and References

1. .
2. Steen & Seebach, example #74
3. Steen & Seebach, example #80