Bulletin of the Australian Mathematical Society

Research Article

NOTE ABOUT LINDELÖF Σ-SPACES υX

J. KA̧KOLa1 and M. LÓPEZ-PELLICERa2 c1

a1 Faculty of Mathematics and Informatics, A Mickiewicz University, 61-614 Poznań, Poland (email: kakol@amu.edu.pl)

a2 Depto. de Matemática Aplicada and IUMPA, Universitat Politècnica de València, E-46022 Valencia, Spain (email: mlopezpe@mat.upv.es)

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.

(Received May 14 2011)

(Online publication September 28 2011)

2010 Mathematics subject classification

  • primary 46A50; secondary 54H05;
  • 28A05

Keywords and phrases

  • countable tightness;
  • K-analytic;
  • Lindelöf Σ-spaces;
  • locally convex spaces;
  • realcompactification;
  • spaces of continuous real-valued maps;
  • web-bounded spaces

Correspondence:

c1 For correspondence; e-mail: mlopezpe@mat.upv.es

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.

Dedicated to the Memory of Professor Klaus D. Bierstedt