Countable compactness of hyperspaces and Ginsburg's problems
by
Jiling Cao
Ehime University, Japan
Coauthors: T. Nogura (Ehime University, Japan), A. H. Tomita (Universidade de São Paulo, Brazil)

All topological spaces here are assumed to be Hausdorff. Given a space X, let 2^X be the space of all nonempty closed subsets of X equipped with the Vietoris topology. A well-known theorem of Vietoris and Michael (Trans. Amer. Math. Soc. 71 (1951), 152-182) asserts that 2^X is compact if and only if X is compact. Motivated by this theorem, it is natural to consider if ``compactness" here can be replaced by either ``countable compactness" or ``pseudocompactness".

In 1975, J. Ginsburg (Canad. J. Math. 27 (1975), 1392-1399) proved the following results: (1) If 2^X is countably compact (pseudocompact Tychonoff), then all finite powers of X are countably compact (pseudocompact Tychonoff); (2) If all powers of X are countably compact, then so is 2^X; (3) There exists a Tychonoff space X all of whose finite powers are countably compact, but whose hyperspace 2^X is not pseudocompact. However, he was unable to solve the following problem:

Problem. Is there any relation between the pseudocompactness (countable compactness) of X^\aleph₀ and that of 2^X?

In this talk, I shall present some recent work around this problem. In particular, the following results are our contributions towards it: (a) For a regular (Tychonoff) homogeneous space X, if 2^X is countably compact (pseudocompact), then X^\aleph₀ is countably compact (pseudocompact); (b) (MA) There exists a Tychonoff space X such that for every cardinal \alpha < 2^c, X^\alpha is countably compact, but 2^X is not countably compact; (c) There exists a countably compact Tychonoff space X such that X^t is countably compact, but 2^X is not countably compact.

Date received: May 19, 2002