Every countable compact subset of $\mathbb{S}^n$ is tame

TL;DR

The paper proves that all countable, compact subsets of spheres in dimensions two and higher are tame, meaning their complements are homeomorphic if the subsets are homeomorphic, thus restricting wild subspaces to contain Cantor sets.

Contribution

It establishes that any two homeomorphic countable, compact subsets of ^n are complemented by homeomorphic spaces, showing all such subsets are tame.

Findings

Countable, compact subsets of ^n are tame.

Homeomorphic subsets have homeomorphic complements.

Wild subspaces must contain a Cantor set.

Abstract

We prove that any two countable, compact, subsets of $\mathbb{S}^n, n\geq 2$ that are homeomorphic also have homeomorphic complements. Thus any wild subspace like the classical construction of Antoine must contain a Cantor set.