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Function spaces on ordinals. (English) Zbl 1121.54031
The main result gives a partial classification of spaces \(C_p([1,\alpha ])\) for ordinals \(\alpha \) with respect to homeomorphisms and uniform homeomorphisms. It extends previous results of S. P. Gul’ko [Bull. Pol. Acad. Sci., Math. 36, 391–396 (1988; Zbl 0754.54010)] and uses some of his methods.
A case which is not covered is whether \(C_p([1,\kappa ^+\cdot \kappa ])\) and \(C_p([1,(\kappa ^+)^2])\) are (uniformly) homeomorphic for cardinals \(\kappa \).
It is recalled that a complete classification of \(C_p([1,\alpha ])\) with respect to linear homeomorphisms was given by J. Baars and J. de Groot [On topological and linear equivalence of certain function spaces. (CWI Tract 86, Stiching Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam) (1992; Zbl 0755.54007)].

54C35 Function spaces in general topology
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