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The 12th Summer Conference on General Topology and its Applications
August 12-16, 1997
Nipissing University
North Bay, ON, Canada

Organizers
Ted Chase, Boguslaw Schreyer, Jodi Sutherland, Murat Tuncali, Stephen Watson

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On resolvability
by
I. Juhász
Mathematical Institute of the Hungarian Academy of Sciences
Coauthors: L. Soukup (Mathematical Institute of the Hungarian Academy of Sciences), Z. Szentmiklóssy (Eötvös Loránd University)

A topological space X is called k-resolvable if it contains k-many pairwise disjoint dense subsets. X is maximally resolvable if it is D(X)-resolvable, where D(X) denotes the minimum size of a non-empty open set in X.

We prove some positive theorems:

  1. If \tight(X) < k=\cf(k) £ D(X) then X is k-resolvable.

  2. Every radial space is maximally resolvable.

  3. If every point in a topological space X is the limit of a non-trivial well-ordered sequence then X is w-resolvable. Consequently every pseudo-radial space is w-resolvable.

We do not know if pseudo-radial spaces are maximally resolvable.

We also construct some counterexamples:

  1. There is a T2 space X such that D(X)=w1, X is w-resolvable but not w1-resolvable.

  2. If CH holds then there is a 0-dimensional T2 space X such that D(X)=w1, X is w-resolvable but not w1-resolvable.

Unfortunately we do not have such a T3 example in ZFC. (ii) gives a partial answer to question 8.10(a) in [].

References

[]
W. W. Comport, S. Garcia-Ferreira, Resolvability: a selective survey and some new results, Top. Appl, 74(1996), 149-167.

Date received: June 30, 1997

Copyright © 1997 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 # caao-51.