Hyperspaces of the double arrow

TL;DR

This paper investigates the homogeneity properties of certain hyperspaces related to the double arrow and Sorgenfrey line, revealing nuanced differences in their topological structures and partially answering an open question in the field.

Contribution

It establishes the non-homogeneity of unions of closed intervals in the double arrow and the homogeneity of convergent sequences in both spaces, advancing understanding of their topological characteristics.

Findings

Unions of at most n closed intervals in the double arrow are not homogeneous.

Spaces of non-trivial convergent sequences in the double arrow and Sorgenfrey line are homogeneous.

The space of closed intervals in the Sorgenfrey line is homogeneous.

Abstract

Let $\mathbb{A}$ and $\mathbb{S}$ denote the double arrow of Alexandroff and the Sorgenfrey line, respectively. We show that for any $n\geq 1$, the space of all unions of at most $n$ closed intervals of $\mathbb{A}$ is not homogeneous. We also prove that the spaces of non-trivial convergent sequences of $\mathbb{A}$ and $\mathbb{S}$ are homogeneous. This partially solves an open question of A. Arhangel'ski\v{i}. In contrast, we show that the space of closed intervals of $\mathbb{S}$ is homogeneous.