Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes
Volume 53 / 2005
Abstract
By $\mathop{\rm Fin}(X)$ (resp. $\mathop{\rm Fin}^k(X)$), we denote the hyperspace of all non-empty finite subsets of $X$ (resp. consisting of at most $k$ points) with the Vietoris topology. Let $\ell_2(\tau)$ be the Hilbert space with weight $\tau$ and $\ell_2^{\rm f}(\tau)$ the linear span of the canonical orthonormal basis of $\ell_2(\tau)$. It is shown that if $E = \ell_2^{\rm f}(\tau)$ or $E$ is an absorbing set in $\ell_2(\tau)$ for one of the absolute Borel classes ${\mathfrak a}_\alpha(\tau)$ and ${\mathfrak M}_\alpha(\tau)$ of weight $\leq \tau$ ($\alpha > 0$) then $\mathop{\rm Fin}(E)$ and each $\mathop{\rm Fin}^k(E)$ are homeomorphic to $E$. More generally, if $X$ is a connected $E$-manifold then $\mathop{\rm Fin}(X)$ is homeomorphic to $E$ and each $\mathop{\rm Fin}^k(X)$ is a connected $E$-manifold.
