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Hilbert space; Hilbert cube; $\Cal F_{\sigma\delta}$-absorber; ambient homeomorphism; function space; $p$-summable sequence
In this paper we consider a number of sequence and function spaces that are known to be homeomorphic to the countable product of the linear space $\sigma$. The spaces we are interested in have a canonical imbedding in both a topological Hilbert space and a Hilbert cube. It turns out that when we consider these spaces as subsets of a Hilbert cube then there is only one topological type. For imbeddings in the countable product of lines there are two types depending on whether the space is contained in a $\sigma$-compactum or not.
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