Homeomorphisms between compact subsets of real numbers
Sławomir Kusiński, Szymon Plewik
TL;DR
The paper addresses the problem of classifying compact subsets of $\mathbb{R}$ up to homeomorphism by reducing topological invariants to countable-order invariants, focusing on tame ($t\mathbb{R}$) sets. It develops the $\mathcal{Q}_Y$ framework of gaps and intervals, uses Cantor’s and Dedekind-completion techniques to realize ordered spaces as embedded in $\mathbb{R}$, and defines a robust notion of tameness via monotone isomorphisms between $\mathcal{Q}_X$ and $\mathcal{Q}_Y$ that preserve interval types. The main contributions include a complete classification framework yielding exactly $\omega_1$ pairwise non-homeomorphic $t\mathbb{R}$-sets, a detailed analysis of the finite-$\mathcal{B}_X$ case, and a transfinite Mazurkiewicz–Sierpiński style construction (via the $S(X)$ operation) showing the existence of at least $\omega_1$ many non-homeomorphic regularly closed compact $t\mathbb{R}$-sets, with CH-independent phenomena and related results for Cantor-like sets. Together, these results establish a concrete reduction of tameness in $\mathbb{R}$ to countable-order invariants and reveal rich, ordinally parametrized hierarchies of compact subsets.
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 $ω_1$ many non-homeomorphic t$\mathbb R$-sets.