Embeddings preserving character and cardinal invariants
by
A. Bella
Catania University
Coauthors: I.V. Yaschenko (Moscow Center for Continuous Mathematical Education)

In 1924, Alexandroff and Uryson asked if every space can be embedded into a H-closed space. In 1930, Tychonoff answered this in the affirmative, and later Katetov proved that every space can be embedded as a dense subset into a H-closed space.

We present a construction of H-closed like extensions of a space which preserves character. This provides counterexamples to the following:

Question A Does the inequality |X| <= 2^\chi(X) hold for every almost Lindelöf Hausdorff space X ?

Question B Does |X| <= 2^\chi(Y) hold whenever X is a H-set of the Hausdorff space Y ?

These questions were motivated by an attempt to generalize the cardinal inequality |X| <= 2^\chi(X), proved for every H-closed space X by A. Dow and J. R. Porter.

In fact, the notion of H-closed space may be weakened either to the notion of almost Lindelöf space - X is almost Lindelöf if every open cover \gamma has a countable subfamily \gamma^\prine such that X = \cup {cl_X(U) : U in \gamma'} - or to the notion of H-set - X is a H-set in Y if every family \gamma of open subsets of Y satisfying X subset \cup \gamma has a finite subfamily \gamma' such that X subset \cup {cl_Y(U) : U in \gamma'}.

Date received: May 31, 1996

Copyright © 1996 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 # caag-05.