The Lindelöf property and fragmentability

Cascales, B.; Namioka, I.; Vera, G.

doi:10.1090/S0002-9939-00-05480-0

The Lindelöf property and fragmentability
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by B. Cascales, I. Namioka and G. Vera PDF

Proc. Amer. Math. Soc. 128 (2000), 3301-3309 Request permission

Abstract:

Let $K$ be a compact Hausdorff space and $C(K)$ the space of continuous real functions on $K$. In this paper we prove that any $t_{p}(K)$-Lindelöf subset of $C(K)$ which is compact for the topology $t_{p}(D)$ of pointwise convergence on a dense subset $D\subset K$ is norm fragmented; i.e., each non-empty subset of it contains a non-empty $t_{p}(D)$-relatively open subset of small supremum norm diameter. Several applications are given.

References

A. V. Arkhangleskii, Problems in $C_p$-theory, Open problems in topology, 601–615, North-Holland, Amsterdam, 1990.
J. Bourgain, D. H. Fremlin, and M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1978), no. 4, 845–886. MR 509077, DOI 10.2307/2373913
Richard D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodým property, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. MR 704815, DOI 10.1007/BFb0069321
B. Cascales, G. Manjabacas, and G. Vera, Fragmentability and compactness in $C(K)$-spaces, Studia Math. 131 (1998), no. 1, 73–87. MR 1628664, DOI 10.4064/sm-131-1-73-87
B. Cascales and G. Vera, Topologies weaker than the weak topology of a Banach space, J. Math. Anal. Appl. 182 (1994), no. 1, 41–68. MR 1265882, DOI 10.1006/jmaa.1994.1066
G. Debs, Points de continuité d’une fonction séparément continue. II, Proc. Amer. Math. Soc. 99 (1987), no. 4, 777–782 (French, with English summary). MR 877056, DOI 10.1090/S0002-9939-1987-0877056-9
Robert Deville, Parties faiblement de Baire dans les espaces de Banach. Applications à la dentabilité et à l’unicité de certains préduaux, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 7, 129–132 (French, with English summary). MR 741077
J. E. Jayne and C. A. Rogers, Borel selectors for upper semicontinuous set-valued maps, Acta Math. 155 (1985), no. 1-2, 41–79. MR 793237, DOI 10.1007/BF02392537
W. Moran, Separate continuity and supports of measures, J. London Math. Soc. 44 (1969), 320–324. MR 236346, DOI 10.1112/jlms/s1-44.1.320
I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515–531. MR 370466, DOI 10.2140/pjm.1974.51.515
I. Namioka, Radon-Nikodým compact spaces and fragmentability, Mathematika 34 (1987), no. 2, 258–281. MR 933504, DOI 10.1112/S0025579300013504
R. Pol, On pointwise and weak topology in function spaces, Preprint Nr. 4/84. Warszawa, 1984.
Haskell P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 358307, DOI 10.1073/pnas.71.6.2411
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
V. V. Srivatsa, Baire class $1$ selectors for upper semicontinuous set-valued maps, Trans. Amer. Math. Soc. 337 (1993), no. 2, 609–624. MR 1140919, DOI 10.1090/S0002-9947-1993-1140919-0
Charles Stegall, Functions of the first Baire class with values in Banach spaces, Proc. Amer. Math. Soc. 111 (1991), no. 4, 981–991. MR 1019283, DOI 10.1090/S0002-9939-1991-1019283-7
Michel Talagrand, Deux généralisations d’un théorème de I. Namioka, Pacific J. Math. 81 (1979), no. 1, 239–251 (French). MR 543747, DOI 10.2140/pjm.1979.81.239
Gabriel Vera, Baire measurability of separately continuous functions, Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 153, 109–116. MR 929799, DOI 10.1093/qmath/39.1.109