Homogeneity and rigidity in Erd\H{o}s spaces

TL;DR

This paper explores the homogeneity properties of Erdős spaces within separable Hilbert spaces, providing examples of non-homogeneous and rigid spaces based on coordinate set constructions.

Contribution

It introduces specific examples of Erdős spaces that are either non-homogeneous or rigid, advancing understanding of their topological structure.

Findings

Provided a non-homogeneous Erdős space example.

Constructed a rigid Erdős space example.

Enhanced understanding of homogeneity in topological subspaces.

Abstract

We investigate the homogeneity of topological subspaces of separable Hilbert space, akin to the spaces with all points rational or all points irrational, so-called Erd\H{o}s spaces. We provide a non-homogeneous example, that is based on one set of coordinates using, and a rigid example, based on a sequence of coordinate sets.