The topology of systems of hyperspaces determined by dimension functions

TL;DR

This paper characterizes the topological structure of hyperspaces of compact subsets of a Peano continuum based on a dimension function, revealing how these structures vary with the dimension parameter.

Contribution

It provides a detailed topological classification of hyperspaces determined by a dimension function on a Peano continuum, extending understanding of their structure.

Findings

Identifies the topological type of hyperspaces based on dimension thresholds.

Describes the system of hyperspaces parametrized by a subset of the dimension range.

Establishes conditions under which the hyperspaces are homeomorphic to known spaces.

Abstract

Given a non-degenerate Peano continuum $X$, a dimension function $D:2^X_*\to[0,\infty]$ defined on the family $2^X_*$ of compact subsets of $X$, and a subset $\Gamma\subset[0,\infty)$, we recognize the topological structure of the system $(2^X,\D_{\le\gamma}(X))_{\alpha\in\Gamma}$, where $2^X$ is the hyperspace of non-empty compact subsets of $X$ and $D_{\le\gamma}(X)$ is the subspace of $2^X$, consisting of non-empty compact subsets $K\subset X$ with $D(K)\le\gamma$.