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NOTE ABOUT LINDELÖF Σ-SPACES υX

Part of: Classical measure theory Topological linear spaces and related structures

Published online by Cambridge University Press:  28 September 2011

J. KA̧KOL  and
M. LÓPEZ-PELLICER
J. KA̧KOL
Affiliation:
Faculty of Mathematics and Informatics, A Mickiewicz University, 61-614 Poznań, Poland (email: [email protected])
M. LÓPEZ-PELLICER*
Affiliation:
Depto. de Matemática Aplicada and IUMPA, Universitat Politècnica de València, E-46022 Valencia, Spain (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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The paper deals with the following problem: characterize Tichonov spaces X whose realcompactification υX is a Lindelöf Σ-space. There are many situations (both in topology and functional analysis) where Lindelöf Σ (even K-analytic) spaces υX appear. For example, if E is a locally convex space in the class 𝔊 in sense of Cascales and Orihuela (𝔊 includes among others (LM ) -spaces and (DF ) -spaces), then υ(E′,σ(E′,E)) is K-analytic and E is web-bounded. This provides a general fact (due to Cascales–Kakol–Saxon): if E∈𝔊, then σ(E′,E) is K-analytic if and only if σ(E′,E) is Lindelöf. We prove a corresponding result for spaces Cp (X) of continuous real-valued maps on X endowed with the pointwise topology: υX is a Lindelöf Σ-space if and only if X is strongly web-bounding if and only if Cp (X) is web-bounded. Hence the weak* dual of Cp (X) is a Lindelöf Σ-space if and only if Cp (X) is web-bounded and has countable tightness. Applications are provided. For example, every E∈𝔊 is covered by a family {Aα :α∈Ω} of bounded sets for some nonempty set Ω⊂ℕℕ.

Keywords

countable tightnessK-analyticLindelöf Σ-spaceslocally convex spacesrealcompactificationspaces of continuous real-valued mapsweb-bounded spaces

MSC classification

Secondary: 46A50: Compactness in topological linear spaces; angelic spaces, etc. 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 1 , February 2012 , pp. 114 - 120
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This research is supported by the project of Ministry of Science and Higher Education, Poland, grant no. N 201 2740 33 and project MTM2008-01502 of the Spanish Ministry of Science and Innovation.

References

[1]Arkhangel’skii, A. V., Topological Function Spaces, Mathematics and its Applications, 78 (Kluwer, Dordrecht, 1992).CrossRefGoogle Scholar
[2]Blasco, J. L., ‘Complete bases in topological spaces II’, Studia Sci. Math. Hungar. 24 (1989), 447–452.Google Scholar
[3]Cascales, B., ‘On K-analytic locally convex spaces’, Arch. Math. (Basel) 49 (1987), 232–244.CrossRefGoogle Scholar
[4]Cascales, B., Ka̧kol, J. and Saxon, S. A., ‘Weight of precompact subsets and tightness’, J. Math. Anal. Appl. 269 (2002), 500–518.CrossRefGoogle Scholar
[5]Cascales, B., Ka̧kol, J. and Saxon, S. A., ‘Metrizability vs. Fréchet–Urysohn property’, Proc. Amer. Math. Soc. 131 (2003), 3623–3631.CrossRefGoogle Scholar
[6]Cascales, B. and Orihuela, J., ‘On compactness in locally convex spaces’, Math. Z. 195 (1987), 365–381.CrossRefGoogle Scholar
[7]Cascales, B. and Orihuela, J., ‘On pointwise and weak compactness in spaces of continuous functions’, Bull. Soc. Math. Belg. (1988), 331–352.Google Scholar
[8]Ferrando, J. C., ‘Some characterization for υX to be Lindelöf or K-analytic in terms of C p(X)’, Topology Appl. 156 (2009), 823–830.CrossRefGoogle Scholar
[9]Ferrando, J. C. and Ka̧kol, J., ‘A note on spaces C p(X) K-analytic-framed in ℝX’, Bull. Aust. Math. Soc. 78 (2008), 141–146.CrossRefGoogle Scholar
[10]Ferrando, J. C., Ka̧kol, J., López Pellicer, M. and Saxon, S. A., ‘Quasi-Suslin weak duals’, J. Math. Anal. Appl. 339 (2008), 1253–1263.CrossRefGoogle Scholar
[11]Ka̧kol, J. and López Pellicer, M., ‘Compact coverings for Baire locally convex spaces’, J. Math. Anal. Appl. 332 (2007), 965–974.CrossRefGoogle Scholar
[12]Nagami, K., ‘Σ-spaces’, Fund. Math. 61 (1969), 169–192.CrossRefGoogle Scholar
[13]Okunev, O. G., ‘On lindelöf σ-spaces of continuous functions in the pointwise topology’, Topology Appl. 49 (1993), 149–166.CrossRefGoogle Scholar
[14]Orihuela, J., ‘Pointwise compactness in spaces of continuous functions’, J. Lond. Math. Soc. (2) 36 (1987), 143–152.CrossRefGoogle Scholar
[15]Valdivia, M., Topics in Locally Convex Spaces, North-Holland Mathematics Studies, 67 (North-Holland, Amsterdam, 1982).Google Scholar
[16]Valdivia, M., ‘Quasi-LB-spaces’, J. Lond. Math. Soc. (2) 35 (1987), 149–168.CrossRefGoogle Scholar