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Let $C$ be a convex, infinite-dimensional, non-locally-compact subset of a separable Frechet space.

If $C$ is a closed subset (or more generally, if $C$ is completely metrizable), then it is known that $C$ is homeomorphic to $\ell^2$, the separable Hilbert space. This is proved in Separable complete ANR's admitting a group structure are Hilbert manifolds by Dobrowolski and Toruńczyk.

In the non-completely-metrizable case, a topological classification of such $C$'s is (apparently) wide-open. I wish to have more examples of such $C$'s. Do they belong to infinitely many homeomorphism types?

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  • $\begingroup$ Let $B\subset C$ be the centrally symmetric open and closed unit balls in $\ell^2$. Do you mean in particular to classify topologically all sets $X$ such that $B\subseteq X\subseteq C$? $\endgroup$
    – Włodzimierz Holsztyński
    Commented Feb 5, 2014 at 5:36
  • $\begingroup$ @WlodzimierzHolsztynski, at least I do not see why such a classification is impossible. As I mentioned, all $G_\delta$ (i.e. all completely metrizable) subsets are homeomorphic. $\endgroup$
    – Igor Belegradek
    Commented Feb 5, 2014 at 12:07

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