Resolvable spaces
by
Lajos Soukup
Kitami Institute of Technology / Alfréd Rényi Institute of Mathematics
Coauthors: István Juhász (Alfréd Rényi Institute of Mathematics), Zoltán Szentmiklóssy (Eötvös University of Budapest)

A topological space X is called \kappa-resolvable if it contains \kappa-many pairwise disjoint dense subsets. Denote by \Delta(X) the minimum of the cardinalities of non-empty open subsets of X.

It is known that if \kappa is a cardinal of cofinality \omega and a topological space X is \lambda-resolvable for each \lambda < \kappa, then it is \kappa-resolvable. We prove that the assumption cf(\kappa)=\omega is unavoidable here: if cf(\kappa) > \omega then there is a 0-dimensional T₂ topological space X such that \Delta(X) >= \kappa, X is \lambda-resolvable for each \lambda < \kappa, but it is not \kappa-resolvable.

We investigate the resolvability of certain special classes of topological spaces, e.g. we prove

1. every monotone normal space without isolated points is \omega-resolvable,
2. every countably compact, regular space without isolated points is \omega₁-resolvable.

Date received: June 6, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caby-09.