Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

Kotaro Mine¹ ; Katsuro Sakai¹ ; Masato Yaguchi¹

¹ Institute of Mathematics University of Tsukuba Tsukuba, 305-8571 Japan

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) no. 4, pp. 409-419.

Résumé

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.

Détail

DOI : 10.4064/ba53-4-6

Keywords: mathop fin resp mathop fin denote hyperspace non empty finite subsets resp consisting points vietoris topology ell tau hilbert space weight tau ell tau linear span canonical orthonormal basis ell tau shown ell tau absorbing set ell tau absolute borel classes mathfrak alpha tau mathfrak alpha tau weight leq tau alpha mathop fin each mathop fin homeomorphic generally connected e manifold mathop fin homeomorphic each mathop fin connected e manifold