Separation of dense subsets
by
Eva Murtinová
Charles University, Prague

A topological space is called \alpha-normal (\beta-normal) if for each pair of its closed disjoint subsets A, B there are open sets U, V such that U contains a dense subset of A, V contains a dense subset of B and U, V are disjoint (have disjoint closures, respectively). A space is alpha-regular if for every point x and a closed set F not containing x there is a neighbourhood U of x and an open set V containing a dense part of F with U, V disjoint.

There are examples showing that these properties do not coincide with each other neither with classical separation axioms.

We mention conditions for \alpha-regular spaces to be regular and show that, in a strong sense, \alpha-regularity is nonproductive.

Date received: June 20, 2002