ABSTRACT
In Definition 4.5, Chapter 2, we called the collection of all open sets r ( d ) of a metric space ( X , d ) the topology induced by a metric. We recall t ha t 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 quality. For instance, equivalent metrics possess the same topology. In addition, through the topology we can 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 tha t preserves convergence and continuity.
