This thesis is a study of a new category Cp. This category has not appeared in the literature as yet The concept of Cp was suggested by my supervisor Dr. L.D. Nel. The results obtained stem from a list ol conjectures provided by him. The introductory chapter is a reference source for subsequent chapters. Chaptei 1 opens with a known larger category serving as backdrop for Cp. A major result is that Cp is a topologicn universe. The second chapter serves as a source of C'p-spaces. The categories of Hausdorff piecompact uniform spaces and Hausdorff Jb-spaces (kT->) are shown to embed fully. The functor G : &T_ —• Cp proves tc preserve finite products. We derive that G "nearly" preserves function spaces. The kernel of Chapter 3 is that CP(A, IR) is reflexive; this is proved by establishing that certain function spaces of Cp and Cc (convergencc spaces) coincide.