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Proceedings of the American Mathematical Society

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On paracompact subsets of linear topological spaces

Author: Richard A. Graff
Journal: Proc. Amer. Math. Soc. 53 (1975), 361-366
MSC: Primary 58B05; Secondary 46A05
MathSciNet review: 0400282
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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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Keywords: Paracompact, sigma-compact, Lindelöf, linear topological space, weak-star, bounded weak-star, infinite dimensional manifold
Article copyright: © Copyright 1975 American Mathematical Society