The 20th Century brought the rise of General Topology. It arose from the
effort to establish a solid base for Analysis and it is intimately
related to the success of set theory. Many Valued Topology and Its
Applications seeks to extend the field by taking the monadic axioms of
general topology seriously and continuing the theory of topological
spaces as topological space objects within an almost completely ordered
monad in a given base category C. The richness of this theory is shown
by the fundamental fact that the category of topological space objects
in a complete and cocomplete (epi, extremal mono)-category C is
topological over C in the sense of J. Adamek, H. Herrlich, and G.E.
Strecker. Moreover, a careful, categorical study of the most important
topological notions and concepts is given - e.g., density, closedness of
extremal subobjects, Hausdorff's separation axiom, regularity, and
compactness. An interpretation of these structures, not only by the
ordinary filter monad, but also by many valued filter monads, underlines
the richness of the explained theory and gives rise to new concrete
concepts of topological spaces - so-called many valued topological
spaces. Hence, many valued topological spaces play a significant role in
various fields of mathematics - e.g., in the theory of locales,
convergence spaces, stochastic processes, and smooth Borel probability
measures.
In its first part, the book develops the necessary categorical basis for
general topology. In the second part, the previously given categorical
concepts are applied to monadic settings determined by many valued
filter monads. The third part comprises various applications of many
valued topologies to probability theory and statistics as well as to
non-classical model theory. These applications illustrate the
significance of many valued topology for further research work in these
important fields.
