Stability and dimension—a counterexample to a conjecture of Chogoshvili

Abstract

For every n ≥ 2 n \geq 2 we construct an n n -dimensional compact subset X X of some Euclidean space E E so that none of the canonical projections of E E on its two-dimensional coordinate subspaces has a stable value when restricted to X X . This refutes a longstanding claim due to Chogoshvili. To obtain this we study the lattice of upper semicontinuous decompositions of X X and in particular its sublattice that consists of monotone decompositions when X X is hereditarily indecomposable.