Homeomorphisms between compact subsets of real numbers

Abstract

We first present a reduction of properties of compact sets of real numbers to properties of countable orders. Then discuss a variant of homogeneity of compact subsets of real numbers, focusing on the family of $t$-sets. Finally, we prove that there are exactly ${\omega }_1$ many non-homeomorphic $t$-sets.

Acknowledgment: We would like to thank W. Bielas and T. Kania for their comments, which allowed us to remove typos and ambiguities from the version previously published at arXiv