Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

Kotaro Mine, Katsuro Sakai, and Masato Yaguchi. "Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes." Bulletin of the Polish Academy of Sciences. Mathematics 53.4 (2005): 409-419. <http://eudml.org/doc/280236>.

@article{KotaroMine2005,
abstract = {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.},
author = {Kotaro Mine, Katsuro Sakai, Masato Yaguchi},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {hyperspace; absolute Borel classes; absorbing set},
language = {eng},
number = {4},
pages = {409-419},
title = {Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes},
url = {http://eudml.org/doc/280236},
volume = {53},
year = {2005},
}

TY - JOUR
AU - Kotaro Mine
AU - Katsuro Sakai
AU - Masato Yaguchi
TI - Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2005
VL - 53
IS - 4
SP - 409
EP - 419
AB - 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.
LA - eng
KW - hyperspace; absolute Borel classes; absorbing set
UR - http://eudml.org/doc/280236
ER -