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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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