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Hausdorff compactifications and zero-one measures - II
by
Gino Tironi
Univ. Trieste, Italy
Coauthors: Georgi D. Dimov (Univ. Sofia, Bulgaria)
Results of our papers [1] and [2] 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 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 BX and a set LX of sublattices of BX 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 LX}? This question is motivated also by some measure-theoretic constructions of Hausdorff compactifications.
The second question is: is it possible to do this for every Hausdorff compactification of X? We shall answer the questions posed above in affirmative. Finally, we will find a necessary and sufficient condition (and also some sufficient conditions) which a L in LX has to satisfy in order to obtain that max(L) is a Wallman compactification of X.
[1] Compactifications, A-Compactifications and Proximities,
Ann. Matem. 169 (1995) 87-108
[2] Hausdorff Compactifications and Zero-One Measures, Math. Proc. Camb. Phil. Soc. (2001) (to appear)
Date received: March 19, 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 # cafw-66.