Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-23T07:38:52.531Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON SPACES Cp(X)K-ANALYTIC-FRAMED IN ℝX

Part of: Connections with other structures, applications Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  01 August 2008

J. C. FERRANDO  and
J. KĄKOL
J. C. FERRANDO*
Affiliation:
Centro de Investigación Operativa, Universidad Miguel Hernández, E-03202 Elche (Alicante), Spain (email: jc.ferrando@umh.es)
J. KĄKOL
Affiliation:
Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland (email: kakol@math.amu.edu.pl)
*
✉For correspondence; e-mail: jc.ferrando@umh.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper characterizes the K-analyticity-framedness in ℝX for Cp(X) (the space of real-valued continuous functions on X with pointwise topology) in terms of Cp(X). This is used to extend Tkachuk’s result about the K-analyticity of spaces Cp(X) and to supplement the Arkhangel ′skiĭ–Calbrix characterization of σ-compact cosmic spaces. A partial answer to an Arkhangel ′skiĭ–Calbrix problem is also provided.

Keywords

P-framed spacecosmic spaceK-analytic spaceLindelöf Σ-spaceangelic space

MSC classification

Secondary: 54C35: Function spaces 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 1 , August 2008 , pp. 141 - 146
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The research of both authors has been supported by project MTM2005-01182 of the Spanish Ministry of Education and Science, co-financed by the European Community (Feder funds). The second named author was also supported by grant MNiSW Nr. N N201 2740 33 as well as by the Technical University of Valencia (September 2007) with the grant ‘Ayuda para estancias de investigadores de prestigio en la UPV’.

References

[1]Arkhangel′skiĭ, A. V., Topological Function Spaces, Math. and its Applications, 78 (Kluwer Academic Publishers, Dordrecht, 1992).Google Scholar
[2]Arkhangel′skiĭ, A. V., ‘A survey on C p-theory’, in: Recent Progress in General Topology (eds. M. Hušek and J. van Mills) (North-Holland, Amsterdam, 1992), pp. 1–48.Google Scholar
[3]Arkhangel′skiĭ, A. V., ‘Function spaces in the topology of pointwise convergence and compact sets’, Russian Math. Surveys 39 (1984), 9–56.Google Scholar
[4]Arkhangel′skiĭ, A. V. and Calbrix, J., ‘A characterization of σ-compactness of a cosmic space X by means of subspaces of ℝX’, Proc. Amer. Math. Soc. 127 (1999), 2497–2504.CrossRefGoogle Scholar
[5]Canela, M. A., ‘Operator and function spaces which are K-analytic’, Port. Math. 42 (1983), 203–218.Google Scholar
[6]Cascales, B., ‘On K-analytic locally convex spaces’, Arch. Math. 49 (1987), 232–244.Google Scholar
[7]Cascales, B. and Orihuela, J., ‘On compactness in locally convex spaces’, Math. Z. 195 (1987), 365–381.Google Scholar
[8]Cascales, B. and Orihuela, J., ‘On pointwise and weak compactness in spaces of continuous functions’, Bull. Soc. Math. Belg. 40 (1988), 331–352.Google Scholar
[9]Cascales, B., Kąkol, J. and Saxon, S. A., ‘Weight of precompact subsets and tightness’, J. Math. Anal. Appl. 269 (2002), 500–518.CrossRefGoogle Scholar
[10]Cascales, B., Kąkol, J. and Saxon, S. A., ‘Metrizability vs. Fréchet–Urysohn property’, Proc. Amer. Math. Soc. 131 (2003), 3623–3631.Google Scholar
[11]Calbrix, J., ‘Espaces K σ et espaces des applications continues’, Bull. Soc. Math. France 113 (1985), 183–203.CrossRefGoogle Scholar
[12]Christensen, J. P. R., Topology and Borel Structure, Mathematical Studies, 10 (North-Holland, Amsterdam, 1974).Google Scholar
[13]Cleary, J. and Morris, S. A., ‘Topologies on locally compact groups’, Bull. Austral. Math. Soc. 38 (1988), 105–111.Google Scholar
[14]Ferrando, J. C., ‘Two new properties of the space C p(X)’, Topology Appl. 154 (2007), 1799–1803.Google Scholar
[15]Ferrando, J. C., Kąkol, J., López Pellicer, M. and Saxon, S. A., ‘Quasi-Souslin weak duals’, J. Math. Anal. Appl. 339 (2008), 1253–1263.CrossRefGoogle Scholar
[16]Floret, K., Weakly Compact Sets, Lecture Notes in Mathematics, 801 (Springer, Berlin, 1980).Google Scholar
[17]Lutzer, D., van Mill, J. and Pol, R., ‘Descriptive complexity of function spaces’, Trans. Amer. Math. Soc. 291 (1985), 121–128.Google Scholar
[18]Okunev, O. G., ‘On Lindelöf σ-spaces of continuous functions in the pointwise topology’, Topology Appl. 49 (1993), 149–166.CrossRefGoogle Scholar
[19]Orihuela, J., ‘Pointwise compactness in spaces of continuous functions’, J. London Math. Soc. 36 (1987), 143–152.CrossRefGoogle Scholar
[20]Talagrand, M., ‘Espaces de Banach faiblement K-analytiques’, Ann. of Math. 110 (1979), 407–438.CrossRefGoogle Scholar
[21]Tkachuk, V. V., ‘A space C p(X) is dominated by irrationals if and only if it is K-analytic’, Acta Math. Hungar. 107(4) (2005), 253–265.Google Scholar
[22]Valdivia, M., Topics in Locally Convex Spaces (North-Holland, Amsterdam, 1982).Google Scholar