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The condition weak-P(g, a) and its usage to count the number of dense w-bounded subgroups.
by
Luis Recoder-Núñez
Wesleyan University
Let \alpha and \gamma be infinite cardinals. Following [1], we say that the condition weak-P(\gamma, \alpha) holds if there is a Hausdorff, zero-dimensional, weak-P-space X of size \gamma and weight at most \alpha.
In [1], Comfort and van Mill prove that condition weak-P(2\alpha, \alpha) holds provided \alpha\omega=\alpha. When G is a compact group which in addition is either Abelian or connected and whose weight is \alpha with \alpha\omega=\alpha, they prove that the number of dense \omega-bounded subgroups of G is 22\alpha. This gives a partial answer to a question posed by Itzkowitz and Shakhmatov in [2].
Regarding condition weak-P(\gamma, \alpha) which does not always hold, we give a partial answer to a question of Comfort and van Mill in [1]. In particular, we prove that if \alpha is a cardinal number with uncountable cofinality, then condition weak-P(\alpha+, \alpha) holds. Using this fact, we prove that if G is as before but with weight \alpha and cf(\alpha) > \omega, then the number of dense \omega-bounded subgroups of G is at least 2(\alpha+). If in addition 2\alpha < 2(\alpha+), then the number of such subgroups is at least |G|+. This also gives a partial answer to the question of Itzkowitz and Shakhmatov mentioned before.
References:
[1] W. W. Comfort and Jan van Mill, How many \omega-bounded subgroups ?, Topology and its applications, 77 (1997), 105-113.
[2] G. Itzkowitz and D. Shakhmatov, Dense countably compact subgroups of compact groups, Math. Japonica, 45 (1997), 497-501.
Date received: May 31, 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-89.