The book explores the interconnectedness of various mathematical disciplines and their applications across different fields, highlighting the evolution of mathematical thought. It emphasizes how seemingly unrelated branches can reveal surprising relationships, showcasing the growing complexity and specialization in mathematics. The author discusses the integration of advanced mathematical concepts in areas such as economics, physics, and engineering, while also addressing the emergence of new subdisciplines that challenge traditional classification.
Convenient Topology introduces a new framework that addresses limitations found in traditional topological and uniform spaces, enhancing the study of convergence and continuity. Central to this approach are semiuniform convergence spaces, which encompass various convergence structures, including topological and uniform types. This framework allows for the exploration of complex concepts such as Cauchy continuity and uniform continuity in function spaces. The book presents unique results unattainable through conventional methods, with a self-contained text, except for the final chapter discussing nearness concepts.
A new foundation of Topology, summarized under the name Convenient Topology, is considered such that several deficiencies of topological and uniform spaces are remedied. This does not mean that these spaces are superfluous. It means exactly that a better framework for handling problems of a topological nature is used. In this setting semiuniform convergence spaces play an essential role. They include not only convergence structures such as topological structures and limit space structures, but also uniform convergence structures such as uniform structures and uniform limit space structures, and they are suitable for studying continuity, Cauchy continuity and uniform continuity as well as convergence structures in function spaces, e.g. simple convergence, continuous convergence and uniform convergence. Various interesting results are presented which cannot be obtained by using topological or uniform spaces in the usual context. The text is self-contained with the exception of the last chapter, where the intuitive concept of nearness is incorporated in Convenient Topology (there exist already excellent expositions on nearness spaces).
An dem für die deutsche Sprache typischen Phänomen der Trennbarkeit vieler Verben entzünden sich seit geraumer Zeit die Gemüter. Wohlbekannt ist dazu das satirische Lamento Mark Twains: „Whenever the literary German dives into a sentence, that is the last you are going to see of him till he emerges on the other side of his Atlantic with his verb in his mouth.“ Während also die einen die sogenannte Verbalklammer als unnötige, verständnishemmende Gedächtnisbelastung verurteilen, unterstellen ihr dagegen andere eine gedächtnisstimulierende Wirkung. In Anbetracht der nachgewiesenen Behaltenswirksamkeit steht weiterhin zu erwarten, dass die Textverständlichkeitsforschung künftig verbale Klammerstrukturen bei der Definition ihrer Textverständlichkeitsdimensionen angemessen berücksichtigen wird.
