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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Dispersed images of topological spaces and uncomplemented subspaces of $ C(X)$


Author: John Warren Baker
Journal: Proc. Amer. Math. Soc. 41 (1973), 309-314
MSC: Primary 54C05; Secondary 46E15
DOI: https://doi.org/10.1090/S0002-9939-1973-0320984-3
MathSciNet review: 0320984
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Abstract: Let $ \Gamma (\alpha )$ denote the set of ordinals not exceeding $ \alpha $ with its interval topology. We show that, if $ X$ is a 0-dimensional Hausdorff space and $ \alpha $ is a denumerable ordinal such that the $ \alpha $th derived set of $ X$ contains $ n$ points where $ n < \omega $, there is a map of $ X$ onto $ \Gamma ({\omega ^\alpha } \cdot n)$. Maps of completely regular spaces into the unit interval are considered and a noncompact analogue of a theorem of Pełczyński and Semadeni is obtained. Finally, these results are used to give a simple proof to the following theorem: If $ X$ is completely regular and $ {X^{(\omega )}} \ne \emptyset $, there is an uncomplemented subspace $ H$ of $ C(X)$ which is isometrically isomorphic to $ C(Y)$ for some compact metric space $ Y$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0320984-3
Keywords: 0-dimensional spaces, dispersed spaces, scattered spaces, ordered topological spaces, Boolean spaces, Banach spaces of continuous functions, uncomplemented subspaces of $ C(X)$
Article copyright: © Copyright 1973 American Mathematical Society