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

Arhangel’skii, A.; Calbrix, J.

doi:10.1090/S0002-9939-99-04782-6

A characterization of $\sigma$-compactness of a cosmic space $X$ by means of subspaces of $R^X$
HTML articles powered by AMS MathViewer

by A. V. Arhangel’skii and J. Calbrix

Proc. Amer. Math. Soc. 127 (1999), 2497-2504

DOI: https://doi.org/10.1090/S0002-9939-99-04782-6

Published electronically: April 15, 1999

PDF | Request permission

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

A. Arhangel′skiĭ, An addition theorem for the weight of sets lying in bicompacts, Dokl. Akad. Nauk SSSR 126 (1959), 239–241 (Russian). MR 0106444
A.V. Arhangel’skii, Function spaces in the topology of pointwise convergence and compact sets, Russian Math. Surveys 39:5 (1984), 9-56.
A. V. Arkhangel′skiĭ, Topological function spaces, Mathematics and its Applications (Soviet Series), vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian by R. A. M. Hoksbergen. MR 1144519, DOI 10.1007/978-94-011-2598-7
A.V. Arhangel’skii, $C_{p}$-theory, in: Recent Progress in General Topology, Hušek, van Mill Ed-rs, Elsevier Sciences Publishers B.V., (1992), 3-48.
Amer Bešlagić, Embedding cosmic spaces in Lusin spaces, Proc. Amer. Math. Soc. 89 (1983), no. 3, 515–518. MR 715877, DOI 10.1090/S0002-9939-1983-0715877-X
Jean Calbrix, Une propriété des espaces topologiques réguliers, images continues d’espaces métrisables séparables, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 2, 81–82 (French, with English summary). MR 676368
Jean Calbrix, Espaces $K_\sigma$ et espaces des applications continues, Bull. Soc. Math. France 113 (1985), no. 2, 183–203 (French, with English summary). MR 820318
Jean Calbrix, Filtres sur les entiers et ensembles analytiques, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 4, 109–111 (French, with English summary). MR 901620
Jean Calbrix, $k$-spaces and Borel filters on the set of integers, Trans. Amer. Math. Soc. 348 (1996), no. 5, 2085–2090 (French, with English summary). MR 1355296, DOI 10.1090/S0002-9947-96-01635-2
Gustave Choquet, Ensembles ${\cal K}$-analytiques et ${\cal K}$-sousliniens. Cas général et cas métrique, Ann. Inst. Fourier (Grenoble) 9 (1959), 75–81 (French). MR 112843
J. P. R. Christensen, Topology and Borel structure, North-Holland Mathematics Studies, Vol. 10, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Descriptive topology and set theory with applications to functional analysis and measure theory. MR 0348724
Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
Z. Frolik, On analytic spaces, Bull. Acad. Polon. Sci., 8, (1961), 747-750.
J. E. Jayne, Structure of analytic Hausdorff spaces, Mathematika 23 (1976), no. 2, 208–211. MR 461466, DOI 10.1112/S0025579300008809
J. E. Jayne and C. A. Rogers, Borel isomorphisms at the first level. I, Mathematika 26 (1979), no. 1, 125–156. MR 557137, DOI 10.1112/S0025579300009712
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.
E. Michael, $\aleph _{0}$-spaces, J. Math. Mech. 15 (1966), 983–1002. MR 0206907
O. G. Okunev, On Lindelöf $\Sigma$-spaces of continuous functions in the pointwise topology, Topology Appl. 49 (1993), no. 2, 149–166. MR 1206222, DOI 10.1016/0166-8641(93)90041-B
O. G. Okunev, Weak topology of a dual space and a $t$-equivalence relation, Mat. Zametki 46 (1989), no. 1, 53–59, 123 (Russian); English transl., Math. Notes 46 (1989), no. 1-2, 534–538 (1990). MR 1019256, DOI 10.1007/BF01159103
Oleg Okunev, On analyticity in cosmic spaces, Comment. Math. Univ. Carolin. 34 (1993), no. 1, 185–190. MR 1240216
Jean Saint-Raymond, Caractérisation d’espaces polonais. D’après des travaux récents de J. P. R. Christensen et D. Preiss, Séminaire Choquet, 11e–12e années (1971–1973), Initiation à l’analyse, Secrétariat Mathématique, Paris, 1973, pp. Exp. No. 5, 10 (French). MR 0473133