The aim of this book is threefold: to reinstate distance functions as a
principal tool of general topology, to promote the use of distance
functions on various mathematical objects and a thinking in terms of
distances also in nontopological contexts, and to make more specific
contributions to distance theory. We start by learning the basic
properties of distance, endowing all kinds of mathematical objects with
a distance function, and studying interesting kinds of mappings between
such objects. This leads to new characterizations of many well-known
types of mappings. Then a suitable notion of distance spaces is
developed, general enough to induce most topological structures, and we
study topological properties of mappings like the concept of strong
uniform continuity. Important results include a new characterization of
the similarity maps between Euclidean spaces, and generalizations of
completion methods and fixed point theorems, most notably of the famous
one by Brouwer. We close with a short study of distance visualization
techniques.