Recognizing 𝜎-manifolds

Henderson, James P.

doi:10.1090/S0002-9939-1985-0792291-4

Recognizing $\sigma$-manifolds
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Proc. Amer. Math. Soc. 94 (1985), 721-727 Request permission

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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Approximating certain cell-like maps by homeomorphisms

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