Factorwise rigidity of embeddings of products of pseudo-arcs

Mauricio E. Chacón-Tirado ¹   ; Alejandro Illanes ¹   ; Rocío Leonel ¹

¹ Instituto de Matematicas Universidad Nacional Autónoma de México Circuito Exterior, Cd. Universitaria México 04510, D.F., México

Colloquium Mathematicum, Tome 128 (2012) no. 1, pp. 7-14

Résumé

An embedding from a Cartesian product of two spaces into the Cartesian product of two spaces is said to be factorwise rigid provided that it is the product of embeddings on the individual factors composed with a permutation of the coordinates. We prove that each embedding of a product of two pseudo-arcs into itself is factorwise rigid. As a consequence, if $X$ and $Y$ are metric continua with the property that each of their nondegenerate proper subcontinua is homeomorphic to the pseudo-arc, then $X\times Y$ is factorwise rigid. This extends results of D. P. Bellamy and J. M. Lysko (for the case that $X$ and $Y$ are pseudo-arcs) and of K. B. Gammon (for the case that $X$ is a pseudo-arc and $Y$ is either a pseudo-circle or a pseudo-solenoid).

Détail

DOI : 10.4064/cm128-1-2

Keywords: embedding cartesian product spaces cartesian product spaces said factorwise rigid provided product embeddings individual factors composed permutation coordinates prove each embedding product pseudo arcs itself factorwise rigid consequence metric continua property each their nondegenerate proper subcontinua homeomorphic pseudo arc times factorwise rigid extends results bellamy lysko pseudo arcs gammon pseudo arc either pseudo circle pseudo solenoid