Topological Spaces

Abstract

Historically, topology has its origin in analysis and geometry and, therefore, some of the basic notions of point-set topology are abstractions of concepts which are classical in the study of analysis. We begin our study of topology by recalling the essential facts about metric spaces. In Sect. 1.2, we shall see that the properties of open sets, closed sets, or neighborhoods of points in a metric space can be used as axioms to introduce a “topology” on a given set. In Sect. 1.3, we discuss certain derived concepts such as “interior”, “closure”, “boundary”, and “limit points” of sets in a topological space. Section 1.4 deals with the notions of “subbase” and “base” for the topology of a space X, which are families of subsets of X that generate the topology by using the operations of intersection and union or the latter one only. In Sect. 1.5, we consider the topology for subsets of a topological space induced by its topology, and study the various notions for subsets of a subspace in relation to the corresponding notion in the given space.