The homeomorphism group of the first uncountable ordinal

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.

Second version: various additions to make a bridge with results of the companion paper arXiv:2011.15009 (new paragraph at the end of the introduction, new Remark 11, new paragraphs in Remarks 18 and 20). Also, new Footnote 2 and new Remark 19. Some results have different labelling or shifted numbering in comparison to the first version