Local Compactness in Approach Spaces Ii

This paper studies the stability properties of the concepts of local compactness introduced by the authors in 1998. We show that all of these concepts are stable for contractive, expansive images and for products. 1. Introduction. Having introduced notions of local compactness, basis local compactness and related measures, and having studied the basic relationship among these concepts in [3], in this paper we study stability properties. Keeping in mind the stability properties of local compactness in TOP, the category of topological spaces and continuous maps, we study how our notions behave under mappings and products. Especially, in the topological case, the maps which are required in order to preserve local compactness are continuous open surjections. This necessitated first finding out what is the right notion of open map in AP. This is duly done in this paper, and it turns out that the notion which we use has nice characterizations for a number of primitive approach structures. Of course, it also turns out to be the right concept in connection with local compactness.