Year: 2019
Type: preprint
Abstract: Hyperspaces $\mathcal H(X)$ of all countable compact subsets of a metric space $X$ and $\mathcal A_n(X)$ of infinite compact subsets which have at most $n$ ($n\in\mathbb N$), or finitely many ($n=ω$) or countably many ($n=ω+1$) accumulation points are studied. By descriptive set-theoretical methods, we fully characterize them for 0-dimensional, dense-in-itself, Polish spaces and partially for $σ$-compact spaces $X$. Using the theory of absorbing sets, we get characterizations of $\mathcal H(X)$, $\mathcal A_ω(X)$ and $\mathcal A_{ω+1}(X)$ for nondegenerate connected, locally connected Polish spaces $X$ which are either locally compact or nowhere locally compact. For every $n\in\mathbb N$, we show that if $X$ is an interval or a simple closed curve, $\mathcal A_n(X)$ is homeomorphic to the linear space $c_{0}=\{(x_{i}) \in\mathbb R^ω: \lim x_{i}=0\}$ with the product topology; if $X$ is a Peano continuum and a point $p\in X$ is of order $\ge 2$, then the hyperspace $\mathcal A_1(X,\{p\})$ of all compacta with exactly one accumulation point $p$ also is homeomorphic to $c_{0}$.
Source: arXiv (Cornell University)
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3163605777