Elements of Point-Set Topology

Abstract

In Definition 4.5, Chapter 2, we called the collection of all open sets τ(d) of a metric space (X, d) the topology induced by a metric. We recall that this collection of open sets or topology is closed with respect to the formation of arbitrary unions and finite intersections. We understand that the topology of a metric space carries the main information about its structural fingerprint. For instance, equivalent metrics possess the same topology. In addition, through the topology we could establish the continuity of a function (see Theorem 4.6, Chapter 2) without need of a metric. This all leads to an idea of defining a structure more general than distance on a set, a structure that preserves convergence and continuity.