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NOTE ABOUT LINDELÖF Σ-SPACES υX

  • J. KA̧KOL (a1) and M. LÓPEZ-PELLICER (a2)
Abstract
Abstract

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⊂ℕ.

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Copyright
Corresponding author
For correspondence; e-mail: mlopezpe@mat.upv.es
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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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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] A. V. Arkhangel’skii , Topological Function Spaces, Mathematics and its Applications, 78 (Kluwer, Dordrecht, 1992).

[3] B. Cascales , ‘On K-analytic locally convex spaces’, Arch. Math. (Basel) 49 (1987), 232244.

[4] B. Cascales , J. Ka̧kol and S. A. Saxon , ‘Weight of precompact subsets and tightness’, J. Math. Anal. Appl. 269 (2002), 500518.

[5] B. Cascales , J. Ka̧kol and S. A. Saxon , ‘Metrizability vs. Fréchet–Urysohn property’, Proc. Amer. Math. Soc. 131 (2003), 36233631.

[6] B. Cascales and J. Orihuela , ‘On compactness in locally convex spaces’, Math. Z. 195 (1987), 365381.

[8] J. C. Ferrando , ‘Some characterization for υX to be Lindelöf or K-analytic in terms of Cp(X)’, Topology Appl. 156 (2009), 823830.

[10] J. C. Ferrando , J. Ka̧kol , M. López Pellicer and S. A. Saxon , ‘Quasi-Suslin weak duals’, J. Math. Anal. Appl. 339 (2008), 12531263.

[11] J. Ka̧kol and M. López Pellicer , ‘Compact coverings for Baire locally convex spaces’, J. Math. Anal. Appl. 332 (2007), 965974.

[13] O. G. Okunev , ‘On lindelöf σ-spaces of continuous functions in the pointwise topology’, Topology Appl. 49 (1993), 149166.

[15] M. Valdivia , Topics in Locally Convex Spaces, North-Holland Mathematics Studies, 67 (North-Holland, Amsterdam, 1982).

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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