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A function space \(C_{p}(X)\) without a condensation onto a \(\sigma\)-compact space. (English) Zbl 1019.54012

Let \(X\) be a completely regular space and let \(C_p(X)\) denote the space of all continuous real-valued functions on \(X\) with the topology of pointwise convergence. We say that a space \(Y\) condenses onto a space \(Z\) in case there is a continuous one-to-one mapping from \(Y\) onto \(Z\). A problem of A. V. Arhangel’skii [Proc. Am. Math. Soc. 128, 1881-1883 (2000; Zbl 0967.54002)] asks whether \(C_p(X)\) condenses onto a \(\sigma\)-compact space in case \(X\) is a separable metric space. The present paper answers this question in the negative with a counterexample of a subset \(X\) of the real line. The construction relies upon the further set-theoretic assumption that the space of irrational numbers can not be covered by less than \(2^\omega\) many compact subsets.

MSC:

54C35 Function spaces in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54G20 Counterexamples in general topology

Citations:

Zbl 0967.54002
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References:

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