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Proceedings of the American Mathematical Society

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Projecting $ C(S)$ onto $ C\sb{0}(S)$


Author: H. Banilower
Journal: Proc. Amer. Math. Soc. 33 (1972), 349-354
MSC: Primary 46E10; Secondary 54C35
DOI: https://doi.org/10.1090/S0002-9939-1972-0295062-1
MathSciNet review: 0295062
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Abstract: If a locally compact Hausdorff space S has a denumerable discrete closed subspace N for which there exists a simultaneous extension E from $ C(N)$ into $ C(S)$ satisfying $ E({C_0}(N)) \subset {C_0}(S)$, then $ {C_0}(S)$ is uncomplemented in $ C(S)$. This holds whenever (i) S is not pseudocompact, or (ii) S is not countably compact and is a subspace of a basically disconnected space.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0295062-1
Keywords: Uncomplemented in $ C(S)$, simultaneous extension, denumerable discrete subspace, basically disconnected, extremally disconnected
Article copyright: © Copyright 1972 American Mathematical Society