On hyperspaces and homeomorphism groups homeomorphic to products of absorbing sets and $R^{\infty}$

Abstract

Two theorems are proven: 1) For a topological space $X$ the pair $(\exp(X), \exp_{\omega}(X))$ of hyperspaces of compact and finite subsets of $X$ is homeomorphic to $(Q\times R^{\infty},\sigma\times R^{\infty})$ if and only if $X$ is a direct limit of a tower $X_{1}\subset X_{2}\subset\cdots$ of strongly countable-dimensional Peano continua such that each $X_{n}$ is nowhere dense in $X_{n+1}$; 2) The triple $(\mathscr{H}^{c}(R), \mathscr{H}_{L^{C}IP}(R),\mathscr{H}_{P^{C}L}(R))$ of homeomorphism groups of the line, endowed with the Whitney topology, is homeomorphic to $(s\times R^{\infty}, \Sigma\times R^{\infty}, \sigma\times R^{\infty})$.