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A characterization of $\sigma $-compactness of a cosmic space $X$ by means of subspaces of $R^{X}$

Author(s): A. V. Arhangel'skii; J. Calbrix
Journal: Proc. Amer. Math. Soc. 127 (1999), 2497-2504.
MSC (1991): Primary 54C35; Secondary 54D45, 28A05.
Posted: April 15, 1999
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Abstract: This work is devoted to the relationship between topological properties of a space $X$ and those of $C_{p}(X)$ (= the space of continuous real-valued functions on $X$, with the topology of pointwise convergence). The emphasis is on $\sigma $-compactness of $X$ and on location of $C_{p}(X)$ in $R^{X}$. In particular, $\sigma $-compact cosmic spaces are characterized in this way.

References:

[1]
A.V. Arhangel'skii, An addition theorem for the weight of sets lying in bicompacta, DAN CCCP, 126, (1959), 239-241. (In Russian.) MR 21:5176

[2]
A.V. Arhangel'skii, Function spaces in the topology of pointwise convergence and compact sets, Russian Math. Surveys 39:5 (1984), 9-56.
[3]
A.V. Arhangel'skii, Topological function spaces, Mathematics and its Applications (Soviet Series), v.78 Kluwer Academic Publishers Group, Dordrecht, 1992. MR 92i:54022
[4]
A.V. Arhangel'skii, $C_{p}$-theory, in: Recent Progress in General Topology, Hu\v{s}ek, van Mill Ed-rs, Elsevier Sciences Publishers B.V., (1992), 3-48.
[5]
A. Be\v{s}lagi\'{c}, Embedding cosmic spaces in Lusin spaces, Proc. Amer. Math. Soc. 89:3 (1983), 515-518. MR 84h:54037
[6]
J. Calbrix, Une propriété des espaces topologiques réguliers, images continues d'espaces métrisables séparables, C. R. Acad. Sc. Paris, t. 295, (1982), 81-82. MR 83j:54029
[7]
J. Calbrix, Espaces $K_{\sigma }$ et espaces des applications continues, Bull. Soc. Math. France, 113, (1985), 183-203. MR 87d:54064
[8]
J. Calbrix, Filtres sur les entiers et ensembles analytiques, C. R. Acad. Sc. Paris, t. 305, (1987), 109-111. MR 88j:54051
[9]
J. Calbrix, k-spaces and Borel filters on the set of integers, Trans. Amer. Math. Soc., 348, 1996, 2085-2090. (In French.) MR 96h:54025
[10]
G. Choquet, Ensembles $K$-analytiques et $K$-Sousliniens, Ann. Inst. Fourier, 9, 1959, 77-81. MR 22:3692a
[11]
J.P.R. Christensen, Topology and Borel Structure, North Holland, Amsterdam, (1974). MR 50:1221
[12]
R. Engelking, General Topology, Heldermann, (1989). MR 91c:54001
[13]
Z. Frolik, On analytic spaces, Bull. Acad. Polon. Sci., 8, (1961), 747-750.
[14]
J.E. Jayne, Structure of analytic Hausdorff spaces, Mathematika 23, (1976), 208-211. MR 57:1451
[15]
J.E. Jayne and C.A. Rogers, Borel isomorphisms at the first level, I, II, Mathematika 26, (1979), 125-179, and 27 (1980), 236-260. MR 81h:54048a; MR 81h:54048b; MR 82e:54043
[16]
C.A. Rogers, J.E. Jayne, C. Dellacherie, F. Topsoe, J. Hoffman-Jorgensen, D.A. Martin, A.S. Kehris, and A.H. Stone, Analytic sets, Academic Press, London, 1980.
[17]
E. Michael, $\aleph _{0}$-Spaces, J. Math. Mech., 15, (1966), 983-1002. MR 34:6723
[18]
O.G. Okunev, On Lindelöf $\Sigma $-spaces of continuous functions in the pointwise topology, Topology and Appl., 49, (1993), 149-166. MR 94b:54055
[19]
O.G. Okunev, Weak topology of an associated space, and $t$-equivalence, Math. Notes 41:1-3 (1990), 534-539. MR 91h:46008
[20]
O.G. Okunev, On analyticity in cosmic spaces, Comment. Math. Univ. Carolinae 34:1 (1993), 185-190. MR 94h:54032
[21]
J. Saint-Raymond, Caractérisation d'espaces polonais, Seminaire d'initiation à l'Analyse (Choquet), 5, (1971-1973), 10 pages. MR 57:12811

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Additional Information:

A. V. Arhangel'skii
Affiliation: January--June: Department of Mathematics, Ohio University, Athens, Ohio 45701 - July--December: Department of Mathematics, Moscow State University, Moscow 119 899, Russia
Email: arhangel@bing.math.ohiou.edu

J. Calbrix
Affiliation: Université de Rouen, URA CNRS 1378, UFR de Sciences, 76821 Mont Saint Aignan Cedex, France
Email: Jean.Calbrix@univ-rouen.fr

DOI: 10.1090/S0002-9939-99-04782-6
PII: S 0002-9939(99)04782-6
Keywords: Function spaces, topology of pointwise convergence, $\sigma $-compactness, $K$-analytic spaces
Received by editor(s): December 8, 1996
Received by editor(s) in revised form: November 1, 1997 and November 12, 1997
Posted: April 15, 1999
Communicated by: Alan Dow
Copyright of article: Copyright 1999, American Mathematical Society

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