Condensations of \(C_p(X)\) onto \(\sigma\)-compact spaces. (English) Zbl 1190.54011

The author answers two questions from [A. V. Arhangel’skii and O. I. Pavlov, Commentat. Math. Univ. Carol. 43, No. 3, 485–492 (2002; Zbl 1090.54003)] by showing that if \(X\) is a non-metrizable Corson compact space then the function space \(C_p(X)\) does not admit a weaker \(\sigma\)-compact topology. The main technical tool is a result on the behaviour of dense subsets of products, that gives sufficient conditions under which continuous images of these have the property that their closed subsets are \(G_\kappa\)-sets. These are in terms of the network weight of the factors and the tightness of the range space. The paper closes with a list of open problems on continuous images and condensations of spaces of the form \(C_p(X)\).
Reviewer: K. P. Hart (Delft)


54C35 Function spaces in general topology
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54D30 Compactness


Zbl 1090.54003
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