Hyperspaces of dimension 1

TL;DR

This paper provides examples of one-dimensional spaces that are not almost zero-dimensional, demonstrating that their hyperspaces of compact subsets also have dimension one, addressing a question about the relationship between space and hyperspace dimensions.

Contribution

It introduces specific examples of one-dimensional, non-almost zero-dimensional spaces with hyperspaces also of dimension one, answering a previously posed question.

Findings

Examples of such spaces are constructed.

Hyperspaces of these spaces are also one-dimensional.

Addresses the relationship between space and hyperspace dimensions.

Abstract

In a previuos paper the author asked if there exists a one-dimensional space $X$ that is not almost zero-dimensional, such that the dimension of the hyperspace of compact subsets of $X$ is one-dimensional. In this short note we give examples of spaces $X$ that are not almost zero-dimensional such that $X$ is one-dimensional and their hyperspace of compacta of $X$ also is one-dimensional.