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On CS-closed sets

Published online by Cambridge University Press:  26 February 2010

David H. Fremlin  and
Michel Talagrand
David H. Fremlin
Affiliation:
UNIVERSITY OF ESSEX, Department of Mathematics, Wivenhoe Park, COLCHESTER CO4 3SQ.
Michel Talagrand
Affiliation:
Equipe d'Analyse, Tour 46, 4 eme étage, Université PARIS VI, 4 Place Jussieu, 75230 PARIS Cedex 05.
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Extract

If C is a convex open set of a Hausdorff locally convex space, then for every compact set K ⊂ C, the closed convex hull of K is contained in C. It follows that the same is true if C is a countable intersection of convex open sets of E. However, as it is well known (and also follows from §7(a)) a convex of Gδ is not always a countable intersection of open convex sets. This paper is devoted to showing that the Gδ convex sets have some remarkable properties.

Type
Research Article
Information
Mathematika , Volume 26 , Issue 1 , June 1979 , pp. 30 - 32
Copyright
Copyright © University College London 1979

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References

1.Jameson, G. J. O.. “Convex series”, Proc. Cambridge Phil. Soc, 72 (1972), 37–47.Google Scholar
2.Fremlin, D. H. and Pryce, J. D.. “Semi-extremal sets and measure representations”, Proc. London Math. Soc. (3), 29 (1974), 502–520.CrossRefGoogle Scholar
3.Weizsäcker, H.. “A note on infinite-dimensional convex sets”, Math. Scand., 38 (1976), 321–324.Google Scholar