Autohomeomorphisms of the finite powers of the double arrow

TL;DR

This paper characterizes homeomorphisms of finite powers of the double arrow space, showing they are locally coordinate permutations composed with monotone embeddings, and investigates homogeneity properties of symmetric products of these spaces.

Contribution

It provides a detailed description of autohomomorphisms of finite powers of the double arrow space and establishes non-homogeneity of their symmetric products for m≥2.

Findings

Homeomorphisms are locally products of monotone embeddings and coordinate permutations.

Symmetric products of the double arrow space are not homogeneous for m≥2.

Symmetric product of 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 any homeomorphism $h:^m\mathbb{A}\to^m\mathbb{A} $ is locally (outside of a nowhere dense set) a product of monotone embeddings $h_i:J_i\subseteq \mathbb{A}\to\mathbb{A} (i\in m)$ followed by a permutation of the coordinates. We also prove that the symmetric products $\mathcal{F}_m(\mathbb{A})$ are not homogeneous for any $m\geq 2$. This partially solves an open question of A. Arhangel'ski\v{i}. In contrast, we show that symmetric product $\mathcal{F}_2(\mathbb{S})$ is homogeneous.