On properties of Rosenthal compacta

Marciszewski, Witold

doi:10.1090/S0002-9939-1992-1096213-4

On properties of Rosenthal compacta
HTML articles powered by AMS MathViewer

by Witold Marciszewski PDF

Proc. Amer. Math. Soc. 115 (1992), 797-805 Request permission

Abstract:

A compact space $K$ is a Rosenthal compactum if $K$ can be embedded in the space ${B_1}(P)$ of the first Baire class functions on $P$, the irrationals, endowed with the pointwise topology. We show that if $L$ is compact, ${C_p}(L)$ (the space of continuous real-valued functions on $L$ with the pointwise topology) is a continuous image of ${C_p}(K)$ and $K$ is a Rosenthal compactum, then $L$ is also. We prove that in some subclass of Rosenthal compacta (compacta consisting of the first Baire class functions with countable supports) the countable chain condition implies separability. We also show that compacta from this class possess a certain covering property hereditarily.

References

K. Alster and R. Pol, On function spaces of compact subspaces of $\Sigma$-products of the real line, Fund. Math. 107 (1980), no. 2, 135–143. MR 584666, DOI 10.4064/fm-107-2-135-143
A. V. Arhangel′skij, Some results and problems in $C_p(X)$-theory, General topology and its relations to modern analysis and algebra, VI (Prague, 1986) Res. Exp. Math., vol. 16, Heldermann, Berlin, 1988, pp. 11–31. MR 952587
S. Argyros and S. Negrepontis, On weakly ${\cal K}$-countably determined spaces of continuous functions, Proc. Amer. Math. Soc. 87 (1983), no. 4, 731–736. MR 687652, DOI 10.1090/S0002-9939-1983-0687652-6
S. Argyros, S. Mercourakis, and S. Negrepontis, Analytic properties of Corson-compact spaces, General topology and its relations to modern analysis and algebra, V (Prague, 1981) Sigma Ser. Pure Math., vol. 3, Heldermann, Berlin, 1983, pp. 12–23. MR 698385
S. Argyros, S. Mercourakis, and S. Negrepontis, Functional-analytic properties of Corson-compact spaces, Studia Math. 89 (1988), no. 3, 197–229. MR 956239, DOI 10.4064/sm-89-3-197-229
J. Bourgain, Some remarks on compact sets of first Baire class, Bull. Soc. Math. Belg. 30 (1978), no. 1, 3–10. MR 549647
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
Dennis K. Burke, Covering properties, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 347–422. MR 776628
W. Wistar Comfort and Stylianos A. Negrepontis, Chain conditions in topology, Cambridge Tracts in Mathematics, vol. 79, Cambridge University Press, Cambridge-New York, 1982. MR 665100
Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
R. J. Gardner, The regularity of Borel measures and Borel measure-compactness, Proc. London Math. Soc. (3) 30 (1975), 95–113. MR 367145, DOI 10.1112/plms/s3-30.1.95
Gilles Godefroy, Compacts de Rosenthal, Pacific J. Math. 91 (1980), no. 2, 293–306 (French, with English summary). MR 615679
Gary Gruenhage, Covering properties on $X^{2}\sbs \Delta$, $W$-sets, and compact subsets of $\Sigma$-products, Topology Appl. 17 (1984), no. 3, 287–304. MR 752278, DOI 10.1016/0166-8641(84)90049-X
Gary Gruenhage, On a Corson compact space of Todorčević, Fund. Math. 126 (1986), no. 3, 261–268. MR 882434, DOI 10.4064/fm-126-3-261-268
Gary Gruenhage, A note on Gul′ko compact spaces, Proc. Amer. Math. Soc. 100 (1987), no. 2, 371–376. MR 884482, DOI 10.1090/S0002-9939-1987-0884482-0
S. P. Gul′ko and T. E. Khmyleva, Compactness is not preserved by the relation of $t$-equivalence, Mat. Zametki 39 (1986), no. 6, 895–903, 943 (Russian). MR 855937
R. Hodel, Cardinal functions. I, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1–61. MR 776620
J. R. Isbell, Mazur’s theorem, General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961) Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962, pp. 221–225. MR 0146640
I. Juhász, Cardinal functions in topology, Mathematical Centre Tracts, No. 34, Mathematisch Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S. Kroonenberg. MR 0340021
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
A. G. Leĭderman, Everywhere dense metrizable subspaces of Corson compacta, Mat. Zametki 38 (1985), no. 3, 440–449, 477 (Russian). MR 811578
Witold Marciszewski, Lindelöf property in function spaces and a related selection theorem, Proc. Amer. Math. Soc. 101 (1987), no. 3, 545–550. MR 908666, DOI 10.1090/S0002-9939-1987-0908666-8
Witold Marciszewski, A function space $C(K)$ not weakly homeomorphic to $C(K)\times C(K)$, Studia Math. 88 (1988), no. 2, 129–137. MR 931038, DOI 10.4064/sm-88-2-129-137
Witold Marciszewski, On a classification of pointwise compact sets of the first Baire class functions, Fund. Math. 133 (1989), no. 3, 195–209. MR 1065902, DOI 10.4064/fm-133-3-195-209
Sophocles Mercourakis, On weakly countably determined Banach spaces, Trans. Amer. Math. Soc. 300 (1987), no. 1, 307–327. MR 871678, DOI 10.1090/S0002-9947-1987-0871678-1
S. Negrepontis, Banach spaces and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1045–1142. MR 776642

On pointwise and weak topology in function spaces

S. Todorčević, Trees and linearly ordered sets, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 235–293. MR 776625
N. N. Yakovlev, On bicompacta in $\Sigma$-products and related spaces, Comment. Math. Univ. Carolin. 21 (1980), no. 2, 263–283. MR 580682