Whitney maps and generalized continua

Abstract

The use of Whitney maps in hyperspace theory is mostly restricted to the class of compact spaces. However, Whitney himself defined these maps in the class of separable metric spaces.Following the original approach of Whitney, we focus on Whitney properties in absence of compactness. We work within the class of generalized continua, and besides classical Whitney maps we will also consider a special kind of such maps termed compactwise proper Whitney maps.In this paper we observe that connectivity properties which are defined piecewise by continua (for instance, continuunwise connectedness) are Whitney properties also in the non-compact setting. These results can be regarded as extensions to generalized continua of well-known results in classical continuum theory.In contrast, ordinary connectedness fails to be a Whitney property in absence of compactness. Notwithstanding, it is a Whitney property for compactwise proper Whitney maps.A relevant feature of non-compact spaces is the connectivity at infinity measured by Freudenthal ends and strong ends. Here we prove that Freudenthal ends are preserved by compactwise proper Whitney maps but not by ordinary Whitney maps and also that the number of strong ends is not a compactwise Whitney property.