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]
Properties and examples
[edit]Every regular space is semiregular.[1] The converse is not true. For example, the space with the double origin topology[2] and the Arens square[3] are Hausdorff semiregular spaces that are not regular.
Open subspaces of a semiregular space are semiregular.[4] But arbitrary subspaces, even closed subspaces, need not be semiregular.[4]
The product of an arbitrary family of semiregular spaces is semiregular.[4]
Every topological space may be embedded into a semiregular space.[1]
See also
[edit]- Separation axiom – Axioms in topology defining notions of "separation"
Notes
[edit]- ^ a b c Willard, Stephen (2004), "14E. Semiregular spaces", General Topology, Dover, p. 98, ISBN 978-0-486-43479-7.
- ^ Steen & Seebach, example #74
- ^ Steen & Seebach, example #80
- ^ a b c Engelking 1989, Problem 2.7.6(b).
References
[edit]- Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4.
- 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.