Path maps and continuity

Abstract

A function f: X-> Y will be called a path map if it maps the paths of X to paths of Y. That is, if u: I->X is a path, then so is f: I-?Y. Any continuous function is a path map. Conversely, I shall show that with mild restrictions on the space X, all path maps f: X-* Y are continuous. A duality consideration yields some more information about the situation. In this case, the object of interest is the following: g: X-> Y is a reverse path map if whenever q: Y->I is a continuous (real-valued) function, then so is qg: X->I. If certain conditions are placed upon Y, then every reverse path map g: X-> Y is continuous. The aim of the following theorems is to reveal some of the interplay between paths, continuous functions, and continuous real-valued functions.