Wild high-dimensional Cantor fences in \(\mathbb{R}^n\). I
type: Item
instance of: scholarly article
title: Wild high-dimensional Cantor fences in \(\mathbb{R}^n\). I
zbMATH Open document ID: 1416.57007
published in: Topology and its Applications
publication date: 2019-04-24T00:00:00Z
review text: Let $C$ denote the Cantor set. The first wild Cantor set the reviewer can trace appeared already in 1921 as Antoine's necklace and was soon followed by the set of bad points of the Alexander horned sphere. Mathematicians have found that in $\mathbb{R}^3$ there exist uncountably many inequivalent embeddings of the Cantor set (see the references in the paper) and that all embeddings of the Cantor set in $\mathbb{R}^2$ are tame. Of course wild embeddings of $C$ appear for all $n\geq 3$. \par $X, Y\subset \mathbb{R}^n$ are equivalently embedded if there is a homeomorphism $h$ of $\mathbb{R}^n$ onto itself such that $h(X)=Y$. In this paper we encounter a more general notion: $X$ can be ambiently embedded in $Y$ if there is a homeomorphism $h$ of $\mathbb{R}^n$ onto itself such that $h(X)\subset Y$. Furthermore, $X, Y\subset \mathbb{R}^n$ are ambiently comparable if at least one of them can be ambiently embedded into the other. The author constructs an embedding $f: C\times C \to \mathbb{R}^n, n\geq 3$, such that for each $x\in C$ the image $f(C\times \{x\})$ is wild. Furthermore, for $x \not= y$, and $x, y \in C$, the Cantor sets $f(C\times \{x\})$ and $f(C\times \{y\})$ are incomparable copies of $C$ in $\mathbb{R}^n$. \par This result is generalized to $X\times C$ for $X\subset \mathbb{R}^{n-1}$ being a non-empty perfect compact space.
reviewed by: Ivan Ivanšić
Mathematics Subject Classification ID: 57M30
Mathematics Subject Classification ID: 54C25
zbMATH DE Number: 7047535
zbMATH Keywords: Euclidean spaces
zbMATH Keywords: equivalent embeddings
zbMATH Keywords: disjoint embeddings
zbMATH Keywords: wild embeddings
zbMATH Keywords: wild Cantor set
author: Olga Frolkina
MaRDI profile type: Publication
full work available at URL: j.topol.2018.09.016
OpenAlex ID: W2923793041
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DOI: 10.1016/J.TOPOL.2018.09.016
description: scientific article; zbMATH DE number 7047535