The pseudocharacter of submaximal spaces
by
Ofelia T. Alas
Universidade de São Paulo
Coauthors: Richard G. Wilson

A crowded space is said to be submaximal if every dense set is open. In their comprehensive article, Arhangel'skii and Collins have asked whether every submaximal space is \sigma-discrete. Schröder has shown that this is not so in the presence of measurable cardinals, but the question remains as to whether every crowded submaximal space X has countable pseudocharacter, that is to say, whether each point of the space is a G_\delta. We give an affirmative answer to this question in the case that (1) \chi(X) <= \omega₁ or (2) \piw(x) <= \omega₁ and in the latter case, we show that X is actually strongly \sigma-discrete. We further show that the statement ``every submaximal space of non-measurable cardinality has countable pseudocharacter" is equivalent to ``every maximal space of non-measurable cardinality is \sigma-discrete"

By noting that each point x of van Douwen's countable regular maximal space X is a remote point of X \{x} in \beta(X \{x}), we construct a separable Tychonoff maximal space of cardinality c, thus answering another problem of Arhangel'skii and Collins.

Date received: February 9, 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 # caak-02.