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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.

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