Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

Kotaro Mine ; Katsuro Sakai ; Masato Yaguchi

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005), p. 409-419 / Harvested from The Polish Digital Mathematics Library

Résumé

By Fin(X) (resp. $Fin^{k}(X)$), 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 $ℓ₂^{f}(τ)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E = ℓ₂^{f}(τ)$ or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes $_α(τ)$ and $_α(τ)$ of weight ≤ τ (α > 0) then Fin(E) and each $Fin^{k}(E)$ are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X) is homeomorphic to E and each $Fin^{k}(X)$ is a connected E-manifold.

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Publié le : 2005-01-01

Zbl 1117.54018

EUDML-ID : urn:eudml:doc:280236