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.

Properties and examples

Every regular space is semiregular. The converse is not true. For example, the space $X={ℝ}^2\cup \{0^{\ast }\}$ with the double origin topology and the Arens square are Hausdorff semiregular spaces that are not regular.

Open subspaces of a semiregular space are semiregular.Template:Sfn But arbitrary subspaces, even closed subspaces, need not be semiregular.Template:Sfn

The product of an arbitrary family of semiregular spaces is semiregular.Template:Sfn

Every topological space may be embedded into a semiregular space.

See also

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References

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