Hyperspaces of infinite compacta with finitely many accumulation points

Paweł Krupski, Krzysztof Omiljanowski

Published 2019-08-07Version 1

Vietoris hyperspaces $\mathcal A_n(X)$ ($\mathcal A_\omega(X)$) of infinite compact subsets of a metric space $X$ which have at most $n$ (finitely many, resp.) accumulation points are studied. If $X$ is a dense-in-itself, 0-dimensional Polish space, then $\mathcal A_n(X)$ is homeomorphic to the product $\mathbb Q^{\mathbb N}$. The hyperspaces $\mathcal A_1(\mathbb Q,\{q\})$ and $\mathcal A_1(\mathbb R\setminus\mathbb Q,\{p\})$ of all $A\in \mathcal A_1(\mathbb Q)$ (resp. $A\in \mathcal A_1(\mathbb R\setminus\mathbb Q)$) which accumulate at $q\in\mathbb Q$ (resp. $p\in \mathbb R\setminus\mathbb Q$) are also homeomorphic to $\mathbb Q^{\mathbb N}$. If $X$ is a nondegenerate locally connected metric continuum then hyperspaces $\mathcal A_n(X)$ are absolute retracts for all $n\in\mathbb N\cup\{\omega\}$. If $X=J=[-1,1]$ or $X=S^1$, the hyperspaces $\mathcal A_n(X)$ are characterized as $F_{\sigma\delta}$-absorbers in hyperspaces $\mathcal K(J)$ and $\mathcal K(S^1)$ of all compacta in $J$ and $S^1$, respectively. Consequently, they are homeomorphic to the linear space $\{(x_k) \in\mathbb R^{\mathbb N}: \lim x_k=0\}$ with the product topology. The hyperspaces $\mathcal A_\omega(X)$ for $X$ being a Euclidean cube, the Hilbert cube, the $m$-dimensional unit sphere $S^m$, $m\ge 1$, or a compact $m$-manifold with boundary in $S^m$, $m\ge 3$, are true $F_{\sigma\delta\sigma}$-sets which are strongly $F_{\sigma\delta}$-universal in the respective hyperspaces of all compacta.