Diagonals Separating the Square of a Continuum

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 $$p,q\in R$$ p , q ∈ R , there exists a subcontinuum M of X such that $$\{p,q\}\subset M\subset R$$ { p , q } ⊂ M ⊂ R . Let $$X^{2}$$ X 2 denote the Cartesian square of X and $$\Delta $$ Δ the diagonal of $$X^{2}$$ X 2 . Recently, H. Katsuura asked if for a continuum X, distinct from the arc, $$X^{2}\setminus \Delta $$ X 2 \ Δ 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 dynamical properties of a suitable homeomorphism of the Cantor set onto itself to construct an appropriate indecomposable continuum X.