Abstract
AbstractA 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.
Funder
CONACyT
National Science Centre, Poland
Publisher
Springer Science and Business Media LLC
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