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On the space of continuous functions on a dyadic set

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  26 February 2010

Grzegorz Plebanek
Grzegorz Plebanek
Affiliation:
Professor Grzegorz Plebanek, Institute of Mathematics, Wroclaw University, Grunwaldzki 2/4, 50384 Wroclaw, Poland.
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Extract

In the part (16-3) of his extensive study on measurability in Banach spaces, Talagrand [12] considered the Banach space C(K) of continuous functions on a dyadic topological space K. He proved that C(K) is realcompact in its weak topology, if, and only if, the topological weight of K is not a twomeasurable cardinal (Theorem 16-3-1). Then he asked for an alternative to a rather complicated proof presented there (p. 214) and posed the problem whether C(K) is measure-compact whenever the weight of K is not a realmeasurable cardinal (Problem 16-3-2).

MSC classification

Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Mathematika , Volume 38 , Issue 1 , June 1991 , pp. 42 - 49
Copyright
Copyright © University College London 1991

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References

1. Antonowskij, M. and Chudnowsky, D.. Some questions of general topology and Tichonov semifields II. Russian Math. Surveys, 31 (1976), 69–128.Google Scholar
2. Ciesielski, K.. Real-valued sequentially continuous functions on the product space 2k. Preprint, not intended for publication.Google Scholar
3. Corson, H. H.. The weak topology of a Banach space. Trans. Amer. Math. Soc, 101 (1961), 1–15.CrossRefGoogle Scholar
4. Edgar, G. A.. Measurability in a Banach space. Indiana Univ. Math. J., 26 (1977), 663–677.CrossRefGoogle Scholar
5. Edgar, G. A.. Measurability in a Banach space II. Indiana Univ. Math. J., 28 (1979), 559–579CrossRefGoogle Scholar
6. Engelking, R.. On functions denned on Cartesian products. Fund. Math., 59 (1966), 221–231.CrossRefGoogle Scholar
7. Engelking, R.. General Topology (PWN Warszawa, 1977).Google Scholar
8. Hagler, J.. On the structure of S and C(S) for S dyadic. Trans. Amer. Math. Soc, 213 (1975), 415–428.Google Scholar
9. Keisler, H. J. and Tarski, A.. From accessible to inaccessible cardinals. Fund. Math., 53 (1964), 225–306.CrossRefGoogle Scholar
10. Mazur, S.. On continuous mappings on Cartesian products. Fund. Math., 39 (1952), 229–238.CrossRefGoogle Scholar
11. Noble, N.. The continuity of functions on Cartesian products. Trans. Amer. Math. Soc, 149 (1970), 187–198.CrossRefGoogle Scholar
12. Talagrand, M.. Pettis integral and measure theory. Mem. Amer. Math. Soc, 51 (1984).Google Scholar
13. Tarski, A. Ideale in vollstandige Mengenkorpen II. Fund. Math., 33 (1945), 51–65.CrossRefGoogle Scholar
14. Wheeler, R. F. A survey of Baire measures and strict topologies. Exp. Math., 2 (1983), 97–190.Google Scholar
15. Linde, W. Probability in Banach Spaces–Stable and Infinitely Divisible Distributions (Wiley … Sons, 1986).Google Scholar