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Proceedings of the American Mathematical Society

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

 

 

On paracompactness in function spaces with the compact-open topology.


Author: Paul O’Meara
Journal: Proc. Amer. Math. Soc. 29 (1971), 183-189
MSC: Primary 54.28
DOI: https://doi.org/10.1090/S0002-9939-1971-0276919-3
MathSciNet review: 0276919
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Abstract: A k-network $ \mathcal{P}$ for a space X is a family of subsets of X such that if $ C \subset U$, with C compact and U open, then there is a finite union R of members of $ \mathcal{P}$ such that $ C \subset R \subset U$. An $ {\aleph _0}$-space is a $ {T_3}$-space having a countable k-network and an $ \aleph $-space is a $ {T_3}$-space having a $ \sigma $-locally finite k-network. In this paper, it is shown that if X is an $ {\aleph _0}$-space and Y is a paracompact $ \aleph $-space, then $ \mathcal{C}(X,Y)$, with the compact-open topology is a paracompact $ \aleph $-space. The result implies that if X is separable metric and Y is metric, then $ \mathcal{C}(X,Y)$ is paracompact.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0276919-3
Keywords: Function spaces, compact-open topology, paracompact, $ \sigma $-locally finite family, k-network, network, $ {\aleph _0}$-space, $ \aleph $-space, $ \sigma $-space, compact-covering mapping, separable metric space, metric space
Article copyright: © Copyright 1971 American Mathematical Society