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On the Complete Regularity of Some Category Spaces

Published online by Cambridge University Press:  20 November 2018

W. Eames
W. Eames*
Affiliation:
City of London Polytechnic, London
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A category space is a measure space which is also a topological space, the measure and the topology being related by ‘a set is measurable iff it has the Baire property’ and ‘a set is null iff it is nowhere dense’ [4]. We considered some category spaces in [3]; now we show that if a null set is deleted from the space, then the topology can be taken to be completely regular. The essential part of the construction consists of obtaining a suitable refinement of the original sequential covering class and using the consequent strong upper density function to define the required topology. Then the complete regularity follows much as in [1].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 5 , February 1975 , pp. 651 - 656
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Chaika, M., The Lusin-Menchoff theorem in metric space, Indiana U. Math. J. 21 (1971), 351–354.Google Scholar
2. Eames, W., A local property of measurable sets, Canad. J. Math. 12 (1960), 632–640.Google Scholar
3. Eames, W., On a topology generated by measurable covers, Canad. Math. Bull. 14 (1971), 499–504.Google Scholar
4. Oxtoby, J. C., Measure and Category, Springer-Verlag, New York, 1971.Google Scholar
5. Zahorski, Z., Sur la premiere derivée, Trans. Amer. Math. Soc. 69 (1950), 1–54.Google Scholar