D-Continuity

Abstract

We call a map f : X → Y D-continuous if its restriction to any set of points that do not possess compact neighborhoods is continuous. We investigate this weaker version of continuity and provide examples to compare D-continuity with other types of continuity. Let f : X → Y be a D-continuous bijective map such that f(A) is locally finite, where A is the set of all points that do not possess compact neighborhoods in X. Then, we show that f|A is a homeomorphism. We also show that if X is a countably generated topological space, then any D-continuous f : X → Y  is continuous. We discuss C-normality, illustrating the relationship between this property and D-continuity. Finally, we investigate the space of all real-valued D-continuous maps of an arbitrary topological space X and obtain some results.