Diagonals separating the square of a continuum

TL;DR

This paper investigates the relationship between the continuumwise connectedness of the square of a continuum minus its diagonal and the decomposability of the continuum, providing counterexamples to a posed question.

Contribution

It demonstrates that the implication between continuumwise connectedness of the square minus the diagonal and decomposability does not hold, using dynamic properties of a homeomorphism of the Cantor set.

Findings

Counterexamples show no equivalence between the properties.

Indecomposable continua can have continuumwise connected squares minus the diagonal.

The question posed in prior work is answered negatively.

Abstract

A metric continuum $X$ is indecomposable if it cannot be put as the union of two of its proper subcontinua. A subset $R$ of $X$ is said to be continuumwise connected provided that for each pair of points $p,q\in R$, there exists a subcontinuum $M$ of $X$ such that $\{p,q\}\subset M\subset R$. Let $X^{2}$ denote the Cartesian square of $X$ and $\Delta$ the diagonal of $X^{2}$. In \cite{ka} it was asked if for a continuum $X$, distinct from the arc, $X^{2}\setminus \Delta$ is continuumwise connected if and only if $X$ is decomposable. In this paper we show that no implication in this question holds. For the proof of the non-necessity, we use the dynamic properties of a suitable homeomorphism of the Cantor set onto itself to construct an appropriate indecomposable continuum $X$.