Proceedings of the American Mathematical Society

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



Recognizing $ \sigma$-manifolds

Author: James P. Henderson
Journal: Proc. Amer. Math. Soc. 94 (1985), 721-727
MSC: Primary 57N20; Secondary 54B15, 54C99, 58B05
MathSciNet review: 792291
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Abstract: Denote by $ \sigma$ the subspace of the Hilbert cube consisting of $ \{ (x_i): x_i = 0$ for all but finitely many $ i$}. Then following characterization of manifolds modeled on $ \sigma $ is proven and applied to cell-like, upper semicontinuous decompositions of $ \sigma $-manifolds. An ANR $ X$ is a $ \sigma $-manifold if and only if (a) $ X$ is the countable union of finite-dimensional compacta, (b) each compact subset of $ X$ is a strong $ Z$-set, and (c) for each positive integer $ k$, every mapping $ f:{R^k} \to X$ can be arbitrarily closely approximated by an injection.

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Keywords: Countable dimensional manifold, cell-like decompositions, Euclidean injection property
Article copyright: © Copyright 1985 American Mathematical Society