Homeomorphisms between compact subsets of real numbers

TL;DR

This paper explores the classification of compact subsets of real numbers through invariants linked to countable orders, introducing tR-sets and establishing their properties and diversity.

Contribution

It introduces the concept of tR-sets, demonstrates their embedding properties, and proves the existence of exactly  many non-homeomorphic tR-sets.

Findings

Existence of exactly  many non-homeomorphic t-sets.

Reduction of compact set properties to countable order invariants.

Embedding properties of certain compact sets called t-sets.

Abstract

A reduction of properties (invariants) of compact sets of real numbers to properties of countable orders is presented here. Discussed here is also an embedding property of some compact sets that are called t$\mathbb R$-sets. Among others, it is proved that there are exactly $\omega_1$ many non-homeomorphic t$\mathbb R$-sets.