II - Topological Spaces

Abstract

The problem of defining a notion of convergence appropriate to any set whatsoever, a notion that lends itself to easy rules of calculation (like the rules for calculating with limits in the classical theory), leads almost inescapably to the concept of topological space. This chapter studies topological spaces from this viewpoint of the problem. The chapter presents a simple and appropriate transcription of specific properties of classical real analysis to define the general concept of topological space. The concept of topological spaces is derivable from a few extra conditions, imposable in a natural way on any notion of convergence. Moreover, it is proved that the second defining property of a topology is an appropriate generalization of the diagonal property of real sequences. Any concept of convergence that verifies conditions (α)–(γ), and in which the generalized diagonal property holds, can be thought of as having been generated by a topology.