# Hyperspaces of countable compacta

## TL;DR

This paper characterizes the topological structure of hyperspaces of countable compact subsets in various metric spaces, revealing homeomorphisms to classical Banach spaces and providing descriptive set-theoretic classifications.

## Contribution

It offers a comprehensive topological and descriptive set-theoretic analysis of hyperspaces of countable compacta, including characterizations for specific classes of spaces and new homeomorphism results.

## Key findings

- Hyperspaces of countable compacta are characterized for 0-dimensional Polish spaces.
- Certain hyperspaces are homeomorphic to the space c0, including those of intervals and simple closed curves.
- The paper provides descriptions of hyperspaces for nondegenerate connected, locally connected Polish spaces.

## 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 ($n=\omega$) or countably many ($n=\omega+1$) accumulation points are studied. By descriptive set-theoretical methods, we fully characterize them for 0-dimensional, dense-in-itself, Polish spaces and partially for $\sigma$-compact spaces $X$. Using the theory of absorbing sets, we get characterizations of $\mathcal H(X)$, $\mathcal A_\omega(X)$ and $\mathcal A_{\omega+1}(X)$ 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 $c_{0}=\{(x_{i}) \in\mathbb R^{\omega}: \lim x_{i}=0\}$ 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 $c_{0}$.