Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T11:31:25.281Z Has data issue: false hasContentIssue false
  • English
  • Français

Generation of σ-algebras, Baire sets and descriptive Borel sets

Part of: Connections with other structures, applications

Published online by Cambridge University Press:  26 February 2010

J. E. Jayne
J. E. Jayne
Affiliation:
Department of Mathematics, University College London.
Get access

Extract

It is well-known that the σ -algebra of Borel subsets of a metric space coincides with the smallest family of sets which contains the open sets and is closed under countable intersections and countable disjoint unions «3, Th.3, p. 348». A deeper and less known result of Sierpiński is that for separable metric spaces the family of open sets may be replaced by the family of closed sets in the above result «16, p. 272–275» (and «17, p. 51» for the real line). This paper gives an in depth analysis of these and related generation processes. Several abstract formulations, generalizations and limiting examples are given.

MSC classification

Secondary: 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Type
Research Article
Information
Mathematika , Volume 24 , Issue 2 , December 1977 , pp. 241 - 256
Copyright
Copyright © University College London 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Frolik, Z.. “A contribution to the descriptive theory of sets”, General Topology and its Relations to Modern Analysis and Algebra I, Proc. Prague Symp., (Academic Press 1962).Google Scholar
2.Frolik, Z.. “A survey of separable descriptive theory of sets and spaces”, Czech. Math. J., 20 (1970), 406–467.CrossRefGoogle Scholar
3.Frolik, Z.. “Analytic and Borelian sets in general spaces”, Proc. London. Math. Soc. (3), 21 (1970), 674–692.CrossRefGoogle Scholar
4.Gordon, H.. “Rings of functions determined by zero-sets”, Pacific J. Math., 36 (1971), 133–157.CrossRefGoogle Scholar
5.Jayne, J. E.. “Characterizations and metrization of proper analytic spaces”, Inventiones Math., 22 (1973), 51–62.CrossRefGoogle Scholar
6.Jayne, J. E.. “Spaces of Baire functions I”, Ann. Inst. Fourier, Grenoble, 24 (1974), 47–76.CrossRefGoogle Scholar
7.Jayne, J. E.. “Generation of Baire sets”, Proc. Prague Symp. on General Topology (1976), 187–194.Google Scholar
8.Kunugui, K.. “Contribution à la théorie des ensembles Boreliens et analytiques, I”, j. Fac. Sci. Hokkaido Imp. Univ., 7 (1938–39), 161–189.Google Scholar
9.Kuratowski, K., Topology, Vol. I (Academic Press, New York, 1966).Google Scholar
10.Kuratowski, K.. Topology, Vol. II, (Academic Press, New York, 1968).Google Scholar
11.Neubrunn, T.. “A note on quantum probability spaces”, Proc. Amer. Math. Soc, 25 (1970), 672–675.CrossRefGoogle Scholar
12.Rogers, C. A.. “Descriptive Borel sets”, Proc. Roy. Soc. A, 286 (1965), 455–478.Google Scholar
13.Rogers, C. A. and Willmott, R. C.. “On the uniformization of sets in topological spaces”, Ada Math., 120 (1968), 1–52.Google Scholar
14.Rogers, C. A.. “Spaces with good Borel structure”, J. London Math. Soc, 2 (1970), 372–384.CrossRefGoogle Scholar
15.Sierpński, W.. General topology (University of Toronto Press, 1952).CrossRefGoogle Scholar
16.Sierpiński, W.. “Le théorème d'unicité de M. Lusin pour les espaces abstraits”, Fund. Math., 21 (1933), 250–275.CrossRefGoogle Scholar
14.Sierpiński, W., Funkeje Przedstawialne Analitycznie (in Polish), (Lwów, 1925).Google Scholar
18.Hausdorff, F.. Set theory. (Chelsea Pub. Co., New York, 1962).Google Scholar