On paracompact subsets of linear topological spaces

Graff, Richard A.

doi:10.1090/S0002-9939-1975-0400282-1

On paracompact subsets of linear topological spaces
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by Richard A. Graff

Proc. Amer. Math. Soc. 53 (1975), 361-366

DOI: https://doi.org/10.1090/S0002-9939-1975-0400282-1

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Abstract:

It is shown that a connected open subset of a $\sigma$-compact topological space is paracompact in the relative topology only if the subspace is $\sigma$-compact. An application is made to demonstrate the existence of nonparacompact open subspaces, in the weak-star and bounded weak-star topologies, of the dual to a nonseparable Banach space. As a corollary, nonempty paracompact manifolds modeled on such a space always have open submanifolds which are not paracompact.

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