On a question of B.J. Baker and M. Laidacker concerning disjoint compacta in $\mathbb R^N$

TL;DR

This paper constructs wild embeddings of polyhedra in Euclidean spaces to address a question about uncountable families of disjoint compacta, using advanced topological techniques and special wild Cantor sets.

Contribution

It provides a novel construction demonstrating the dual possible answers to a longstanding question about disjoint compacta in Euclidean spaces.

Findings

Wild embeddings can produce uncountable families of disjoint compacta.

Use of Antoine--Blankinship--Ivanov necklaces and Krushkal sticky sets.

Application of Antoine's methods and Shtan'ko dimension theory.

Abstract

We describe wild embeddings of polyhedra into $\mathbb{R}^N$ which show that the answer to the question of B.J. Baker--M. Laidacker (1989) concerning uncountable families of pairwise disjoint compacta can be twofold. The central idea of our construction is the use of specific wild Cantor sets, namely, Antoine--Blankinship--Ivanov necklaces and Krushkal sticky sets. Our basic tools are Antoine's methods and Shtan'ko demension theory.