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Homomorphism-Compact Spaces

Published online by Cambridge University Press:  20 November 2018

A. G. A. G. Babiker  and
S. Graf
A. G. A. G. Babiker
Affiliation:
University of Khartoum, Khartoum, Sudan
S. Graf
Affiliation:
Universität Erlangen-Nürnberg, Erlangen, West Germany
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In 1979 Edgar asked for a characterization of those completely regular Hausdorff topological spaces X which have the property that any Boolean σ-homomorphism from the Baire σ-field of X into the measure algebra of an arbitrary complete probability space can be realized by a measurable point-mapping. Those spaces X will be called homomorphism-compact or, for short, H-compact hereafter. It is wellknown that compact spaces are H-compact (cf. [4], p. 637, Proposition 3.4). We will show that the same is true for strongly measure compact spaces. On the other hand H-compact spaces are easily seen to be real-compact. Since the notions of measure-compactness and liftingcompactness (cf. [3]) also lie between strong measure-compactness and real-compactness it is natural to investigate the relations among these notions. Here the results are mainly negative (cf. Sections 4 and 6). Concerning the structural properties of H-compactness not very much can be said so far (cf. Section 7): it is, for instance, unknown whether the product of two H-compact spaces is again H-compact.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 3 , 01 June 1983 , pp. 558 - 576
Copyright
Copyright © Canadian Mathematical Society 1983

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