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Transactions of the American Mathematical Society

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On the existence of compact metric subspaces with applications to the complementation of $ c\sb{0}$


Authors: William H. Chapman and Daniel J. Randtke
Journal: Trans. Amer. Math. Soc. 219 (1976), 133-148
MSC: Primary 54E45; Secondary 46A99
DOI: https://doi.org/10.1090/S0002-9947-1976-0410688-8
MathSciNet review: 0410688
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Abstract: A topological space X has property $ \sigma - {\text{CM}}$ if for every countable family F of continuous scalar valued functions on X there is a compact metrizable subspace M of X such that $ f(X) = f(M)$ for every f in F. Every compact metric space, every weakly compact subset of a Banach space and every closed ordinal space has property $ \sigma - {\text{CM}}$. Every continuous image of an arbitrary product of spaces having property $ \sigma - {\text{CM}}$ also has property $ \sigma - {\text{CM}}$. If X has property $ \sigma - {\text{CM}}$, then every copy of $ {c_0}$ in $ C(X)$ is complemented in $ C(X)$. If a locally convex space E belongs to the variety of locally convex spaces generated by the weakly compactly generated Banach spaces, then every copy of $ {c_0}$ in E is complemented in E.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0410688-8
Keywords: Metric space, dyadic space, ordinal space, weakly compact subset of a Banach space, weakly compactly generated Banach space, product space, $ \Sigma $-product space, sequential limit point, complemented copy of $ {c_0}$
Article copyright: © Copyright 1976 American Mathematical Society

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