Chapter 14 Categorical Topology

Abstract

Publisher Summary This chapter discusses the first steps of categorical topology and defines categories and functors. It explains the category of all topological spaces and the factorization structures of categories and general treatments. The chapter introduces the notion of reflective subcategory, characterization theorem, diagram and limit, which include the notion of product, and a new notion that induces both separation axioms and connectedness. The characterization theorem of epireflective subcategories in which several standard methods in categorical topology are given is discussed in the chapter. The characterization theorem is a categorical version of the theory of Stone–Cech compactification. The chapter deals with the classical problem concerning the special epireflective subcategories of Top and presents the structure of the collection of all epireflective subcategories of Top. In recent developments of categorical topology, the investigations of topology functors and topological categories are most important. The chapter concludes with the definitions of topology functors and topological categories and several fundamental results.