ABSTRACT
This chapter presents the axiomatic abstract framework of topological spaces. The axioms defining a topology and the concepts associated to it were developed as the result of an arduous and lengthy endeavour during the 19th and early 20th century to isolate and abstract the underlying ideas related to limiting processes. The considerations suggest that distance functions are auxiliary to the definition of convergence and continuity, since the authors merely use a distance function to produce neighbourhoods and open sets. The definition of a topological space relies solely upon set theory and is the most general concept of a mathematical space allowing for analytical notions such as convergence and continuity to make sense. The chapter discusses the notion of convergence of sequences in general topological spaces, as well as the concept of continuity of a mapping between two topological spaces together with their main properties.
