Hausdorff Compactifications and Zero-one Measures
by
Gino Tironi
Department of Math Sciences, Univ of Trieste, via Alfonso Valerio 12/1, I-34127 Trieste, Italy
Coauthors: Georgi D. Dimov (Sofia)

Results of our papers (``Compactifications, A-Compactifications and Proximities". Ann. Matem. 169 (1995), 87-108; and ``Hausdorff Compactifications and Zero-One Measures". Math. Proc. Camb. Phil. Soc. (2001) (to appear)) will be presented in a new simplified form.

In 1964, O. Frink introduced the concept of Wallman-type compactification (or simply ``Wallman compactification") and posed the question of whether each Hausdorff compactification of a Tychonoff space X is a Wallman-type compactification. In 1977, V. M. Ul'janov obtained a negative answer to Frink's question.

A natural question arises: Is it possible to correlate (in a canonical way) to each Tychonoff space X a Boolean algebra B_X and a set L_X of sublattices of B_X in order to obtain that the set of all, up to equivalence, Hausdorff compactifications of X is represented by the set {max(L): L in L_X}? This question is motivated also by some measure-theoretic constructions of Hausdorff compactifications. In fact, Wallman compactifications can also be obtained as a space of regular zero-one measures on a Boolean algebra.

The second question is: Is it possible that every Hausdorff compactification of X be represented as a space of zero-one measures?

We shall answer the questions posed above in affirmative. Finally, we will find a necessary and sufficient condition (and also some sufficient conditions) which an L in L_X has to satisfy in order to obtain that max(L) is a Wallman compactification of X.

Date received: May 16, 2001

Copyright © 2001 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 # cagw-55.