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function space; topology of pointwise convergence
Arhangel'ski\v{\i} proved that if $X$ and $Y$ are completely regular spaces such that ${C_p (X)}$ and ${C_p (Y)}$ are linearly homeomorphic, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. In addition he proved the same result for compactness, $\sigma $-compactness and realcompactness. In this paper we prove that if $\phi : {C_p (X)} \rightarrow {C_p (X)}$ is a continuous linear surjection, then $Y$ is pseudocompact provided $X$ is and if $\phi $ is a continuous linear injection, then $X$ is pseudocompact provided $Y$ is. We also give examples that both statements do not hold for compactness, $\sigma $-compactness and realcompactness.
[1] Arhangel'skiĭ A.V.: On linear homeomorphisms of function spaces. Soviet Math. Dokl. 25 (1982), 852-855.
[2] Baars J., de Groot J.: On Topological and Linear Equivalence of Certain Function Spaces. CWI-tract 86, Centre for Mathematics and Computer Science, Amsterdam. MR 1182148 | Zbl 0755.54007
[3] Baars J., de Groot J., Pelant J.: Function spaces of completely metrizable spaces. to appear in Trans. of the AMS. MR 1182148 | Zbl 0841.54012
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