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The space of continuous functions of a countable metric space. (L’espace des fonctions continues d’un espace métrique dénombrable.) (French) Zbl 0735.54008
If \(X\) is a space then \(C_ p(X)\) denotes the subspace of \(\mathbb{R}^ X\) consisting of all continuous functions. In this interesting paper the author proves that if \(X\) is any countable infinite metrizable space which is not discrete then \(C_ p(X)\) is homeomorphic to \(\sigma_ \omega\). Here \(\sigma_ \omega\) denotes the countable infinite product of copies of \(\ell^ 2_ f\), the subspace of Hilbert space \(\ell^ 2\) consisting of all sequences that are eventually zero. The same result was proved independently by T. Dobrowolski, S. P. Gul’ko and J. Mogilski in Topology Appl. 34, No. 2, 153-160 (1990; Zbl 0691.57009).
Reviewer: J.van Mill

MSC:
54C35 Function spaces in general topology
57N17 Topology of topological vector spaces
58B05 Homotopy and topological questions for infinite-dimensional manifolds
57N20 Topology of infinite-dimensional manifolds
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