Distance and Topology- Back to the Roots of Topology with a General Concept of Distance

The aim of this book is threefold: to reinstate distance functions as a principaltool of general topology to promote the use of distance functions on variousmathematical objects and a thinking in terms of distances also in nontopologicalcontexts and to make more specific contributions to distancetheory.We start by learning the basic properties of distance endowing all kinds ofmathematical objects with a distance function and studying interesting kindsof mappings between such objects. This leads to new characterizations ofmany well-known types of mappings. Then a suitable notion of distancespaces is developed general enough to induce most topological structuresand we study topological properties of mappings like the concept of stronguniform continuity.Important results include a new characterization of the similarity mapsbetween Euclidean spaces and generalizations of completion methods andfixed point theorems most notably of the famous one by Brouwer. We closewith a short study of distance visualization techniques.