Factorwise rigidity of products of pseudo-arcs (Q1085477)

If \(X=\prod_{\gamma \in \Gamma}X_{\gamma}\) and H is a homeomorphism on X such that there exist a one-to-one map \(\phi\) of \(\Gamma\) onto \(\Gamma\) and for each \(\gamma\) in \(\Gamma\) a homeomorphism \(h_{\gamma}\) of \(X_{\gamma}\) onto \(X_{\phi (\gamma)}\) such that \(H(x)_{\phi (\gamma)}=h_{\gamma}(x_{\gamma})\) for each x in X, then H is said to be factor preserving. If every homeomorphism on X is factor preserving, then X is said to be factorwise rigid. It is proved that every product of pseudoarcs is factorwise rigid. The proof involves many known properties of the pseudoarc, including homogeneity. The authors ask whether every product of hereditarily indecomposable continua is factorwise rigid.