In 1915 A. Denjoy has introduced the term of approximately continuous functions which have many nice properties. For example: Every approximately continuous function has the Darboux property and belongs to the first class of Baire; any bounded approximately continuous function is a derivative.
Almost half a century later, a new topology - the so-called density topology - on the real line was introduced. This topology has many ”bad” properties. For example: The density topology is not normal; only the finite sets are density compact.
But it was proved that the ”nice” system of all approximately continuous functions is exactly the class of all functions continuous in the ”bad” density topology.
On the other hand, in 1939 M. Brelot has introduced the notion of thinness of a set in potential theory and very early H. Cartan observed that a set is thin at a point if and only if the complement is a neighborhood of the point in the coarsest topology on \(R^ n\) making all superharmonic functions continuous. This new topology was termed the fine topology and it is known that if \(n>1\) then (for example): The fine topology is not normal; the only fine compact sets are the finite sets; finely continuous functions are of the first class of Baire.
It is seen that the density and fine topologies have many fundamental properties in common. Besides, certain methods and procedures used in connection with the density topology have recently influenced the investigations of fine topology in potential theory.
The present book is based on the lucky connection of two seminars established on the Charles University Prague - the seminar on potential theory and the seminar on modern theory of real functions. Both the fundamenta concepts - the density topology in the real analysis and the fine topology in the potential theory - are studied together under the name ”abstract fine topology”. From this point of view this theory can be considered as a part of the general theory of bitopological spaces.
The first part of the book is devoted to the abstract fine topology. One of the most general properties of the fine topologies is the Lusin- Menchoff property (”binormality”). The fundamental technical means for constructing various functions is the ”abstract between theorem”. Some others problems investigated in this part are the following: Under which circumstances a space with a fine topology is a Baire space, a strong Baire space or a Blumberg space; when finely continuous functions are in the Baire class one; the problems of fine limits and of connectedness of fine topologies.
In the second part - fine topology methods in real analysis - some fine topologies that occur in real analysis are investigated and the general theorems produced in the first part are used in high degree. The fundamental example of fine topology is, of course, the density topology which is studied in various degrees of generality - from the density topology on the line to the abstract density topology investigated in lifting theory. There are found some simple conditions under which the density topology has the Lusin-Menchoff property and the internal characterization of abstract density topologies. The Lusin-Menchoff property is applied to obtain results concerning constructions of various functions, especially approximately continuous functions, where many of the results are new.
The final part - fine topology methods in potential theory - is devoted to fine potential theory. A mong the purely topological properties of harmonic spaces the Lusin-Menchoff property of fine topology is of major importance and is applied in various connections. The theory of finely hyperharmonic functions and of fine Dirichlet problem in harmonic spaces is presented. It is shown a quite new approach to the solution of the fine Dirichlet problem which is based on quasitopological concepts. The quasitopological methods are also used to characterize finely hyperharmonic functions. The possibility of generalization of the presented results to standard H-cones is suggested in the Appendix.
Each section (14 sections) ends with ”Exercises” containing not only counterexamples and generalizations but also further new results. For instance some of results of the theory of bitopological spaces, further generalizations of abstract density topologies, the study of Keldych operators on finely open sets, boundary behaviour of the Perron solution etc. are served as an example.
The ”Exercises” are followed by ”Remarks and comments”. In those excellent remarks the authors give informations concerning the origin and the historical development of the concepts and results of the section and bring further quotations.
It can be expected that this voluminous book will have non-zero influence in both the real analysis and the potential theory.