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Characterizing the topology of infinite-dimensional $ \sigma $-compact manifolds


Author: Jerzy Mogilski
Journal: Proc. Amer. Math. Soc. 92 (1984), 111-118
MSC: Primary 57N20; Secondary 54C55
DOI: https://doi.org/10.1090/S0002-9939-1984-0749902-8
MathSciNet review: 749902
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Abstract: A metric space $ (X,d)$, which is a countable union of finite-dimensional compacta, is a manifold modelled on the space $ l_2^f = \{ ({x_i}) \in {l_2}$ all but finitely many $ {x_i} = 0\} $ iff $ X$ is an ANR and the following condition holds: given $ \varepsilon > 0$, a pair of finite-dimensional compacta $ (A,B)$ and a map $ f:A \to X$ such that $ f\vert B$ is an embedding, there is an embedding $ g:A \to X$ such that $ g\left\vert {B = f} \right\vert B$ and $ d(f(x),g(x)) < \varepsilon $ for all $ x \in A$. An analogous condition characterizes manifolds modelled on the space $ \Sigma = \{ ({x_i}) \in {l_2}:\sum _{i = 1}^\infty {(i{x_i})^2} < \infty \} $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0749902-8
Keywords: Sigma-compact metric ANR's, the strong universality property for compacta, infinite-dimensional sigma-compact manifolds, $ Z$-sets, near-homeomorphisms
Article copyright: © Copyright 1984 American Mathematical Society

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