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A theorem on function spaces

Authors: Jan Baars, Joost de Groot and Jan van Mill
Journal: Proc. Amer. Math. Soc. 105 (1989), 1020-1024
MSC: Primary 54C35; Secondary 54A25
MathSciNet review: 943792
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Abstract: Let $ X$ and $ Y$ be normal and first countable spaces, such that $ {C_p}(X)$ and $ {C_p}(Y)$ are linearly homeomorphic. Suppose $ {X^{(\alpha )}}$ is countably compact for some $ \alpha < {\omega _1}$. We prove that if $ \alpha = 1$ then $ {Y^{(\alpha )}}$ is also countably compact. The first countability condition in this result is essential. We also present examples that if $ \alpha $ is not a prime component, then $ {Y^{(\alpha )}}$ need not to be countably compact.

References [Enhancements On Off] (What's this?)

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Keywords: Function spaces
Article copyright: © Copyright 1989 American Mathematical Society