Semiregular space

From HandWiki

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 [math]\displaystyle{ X = \Reals^2 \cup \{0^*\} }[/math] with the double origin topology[2] and the Arens square[3] are examples of spaces that are Hausdorff semiregular, but not regular.

See also

Notes

  1. 1.0 1.1 Willard, Stephen (2004), "14E. Semiregular spaces", General Topology, Dover, p. 98, ISBN 978-0-486-43479-7, https://books.google.com/books?id=-o8xJQ7Ag2cC&pg=PA98 .
  2. Steen & Seebach, example #74
  3. Steen & Seebach, example #80

References