Watts Spaces and Smooth Maps

Abstract

A new category is described, which generalizes a select variety of categories having smooth mappings as their class of morphisms, those like the C^∞ manifolds and C^∞ mappings between them. Categorical embeddings are produced to justify this claim of generalization. Theorems concerning the equivalence of smooth and continuous versions of different separation axioms are proved following the categorical discussion. These are followed by a generalization of Whitney's approximation theorem, a smooth version of the Tietze extension theorem, and a sufficient condition to guarantee that the connected components of these spaces are smoothly path-connected.