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.
[.]
The space
with the
double origin topology and the
Arens square
In mathematics, the Arens square is a topological space, named for Richard Friederich Arens. Its role is mainly to serve as a counterexample.
Definition
The Arens square is the topological space (X,\tau), where
:X=((0,1)^2\cap\mathbb^2)\cup\\cup ...
[Steen & Seebach, example #80] are examples of spaces that are
Hausdorff semiregular, but not regular.
See also
*
Notes
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).
*
Properties of topological spaces
Separation axioms