Some applications of the topological characterizations of the sigma-compact spaces 𝑙²_{𝑓} and Σ

Curtis, Doug; Dobrowolski, Tadeusz; Mogilski, Jerzy

doi:10.1090/S0002-9947-1984-0743748-7

Some applications of the topological characterizations of the sigma-compact spaces $l^{2}_{f}$ and $\Sigma$
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by Doug Curtis, Tadeusz Dobrowolski and Jerzy Mogilski PDF

Trans. Amer. Math. Soc. 284 (1984), 837-846 Request permission

Abstract:

We use a technique involving skeletoids in $\sigma$-compact metric $\text {ARs}$ to obtain some new examples of spaces homeomorphic to the $\sigma$-compact linear spaces $l_f^2$ and $\Sigma$. For example, we show that (1) every ${\aleph _0}$-dimensional metric linear space is homeomorphic to $l_f^2$; (2) every $\sigma$-compact metric linear space which is an $\text {AR}$ and which contains an infinite-dimensional compact convex subset is homeomorphic to $\Sigma$; and (3) every weak product of a sequence of $\sigma$-compact metric $\text {ARs}$ which contain Hilbert cubes is homeomorphic to $\Sigma$.

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