Openness of the induced map cn(f)

Given a map between compact metric spaces f : X-- and gt;Y , it is possible to induce a map between the n-fold hyperspaces Cn(f) : Cn(X) -- and gt; Cn(Y ) for each positive integer n. Let A and B be classes of maps. A general problem is to find the interrelations between the following two statements:1. f 2 A; 2. Cn(f) 2 B. It is known that 1 and 2 are equivalentconditions if both A and B are the class of homeomorphisms. If A and B are the class of open maps, then the openness of Cn(f) implies the openness of f. Furthermore, there exists an open map f such that Cn(f) is not open. Moreover, if Cn(f) is open and n and gt; 3, then f is both open and monotone. Our main result is Theorem 3.2, where we prove that if the induced map Cn(f) is an open map, for n and gt;= 2, then f is a homeomorphism.

• Autores: Camargo, Javier
• Tipo de recurso: Article of journal
• Fecha de publicación: 2009
• Institución: Universidad Nacional de Colombia
• Repositorio: Universidad Nacional de Colombia
• Idioma: spa
• OAI Identifier: oai:repositorio.unal.edu.co:unal/73768
• Acceso en línea: https://repositorio.unal.edu.co/handle/unal/73768; http://bdigital.unal.edu.co/38245/
• Palabra clave: continua; hyperspaces of continua; induced maps; open maps.
• Rights: openAccess
• License: Atribución-NoComercial 4.0 Internacional