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Hyperspaces of finite sets in universal spaces for absolute Borel classes

Mine K., Sakai K., Yaguchi M.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 4

409-419

By Fin(X) (resp. Fink (X)), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ2 (τ) be the Hilbert space with weight τ and ℓf2 (τ) the linear span of the canonical orthonormal basis of ℓ2 (τ). It is shown that if E = ℓf2 (τ) or E is an absorbing set in ℓ2 (τ) for one of the absolute Borel classes aα (τ) and Mα (τ) of weight ≤ τ (α > 0) then Fin(E) and each Fink (E) are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X) is homeomorphic to E and each Fink (X) is a connected E-manifold.