Extensions of continuous functions: reflective functors

Abstract

The purpose of this paper is to develop a general technique for attacking problems involving extensions of continuous functions from dense subspaces and to use it to obtain new results as well as to improve some of the known ones. The theory of structures developed by Harris is used to get some general results relating filters and covers. A necessary condition is derived for a continuous functionf:X→Yto have a continuous extensionf̅: λx→ λywhere λZdenotes a given extension of the spaceZ. In the case of simple extensions,f̅is continuous and in the case of strict extensionsf̅is θ-continuous. In the case of strict extensions, sufficient conditions for uniqueness off̅are derived. These results are then applied to several extensions considered by Banaschewski, Fomin, Katĕtov, Liu-Strecker, Blaszczyk-Mioduszewski, Rudölf, etc.