The homeomorphism group of the first uncountable ordinal

TL;DR

This paper investigates the topological properties of the homeomorphism group of the first uncountable ordinal, demonstrating compatibility with pointwise convergence and establishing properties like amenability and Roelcke-precompactness.

Contribution

It introduces new topological insights into the homeomorphism group of the first uncountable ordinal, including compatibility with pointwise convergence and specific properties.

Findings

Homeomorphism group is compatible with pointwise convergence.

The group exhibits amenability.

The group is Roelcke-precompact.

Abstract

We show that the topology of pointwise convergence on scattered spaces is compatible with the group structure of their homeomorphism group. We then establish a few topological properties of the homeomorphism group of the first uncountable ordinal, such as amenability and Roelcke-precompactness.