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Convergence Theory


(see "An initiation into convergence theory"  pdf   52 pages)

    The role of convergence theory with respect to topology is similar to that of complex numbers with regard to real numbers. The field of complex numbers is the least algebraically complete field that includes the reals. Actually the category of convergences is closed for the important operation of forming powers, but the least subcategory of convergences closed for initial convergences and powers, and containing all the topologies, is that of epitopologies.
    Many problems formulated in terms of topologies have no solutions within the class of topologies, but have within that of convergences. For instance (in general) there is no coarsest topology on the set of continuous functions for which the evaluation map is continuous; but there exists the coarsest convergence (In terms of category theory, the category of topologies is not cartesian closed, but that of convergences is). On the other hand, numerous classical properties in topology, coincide with the classes of solutions of certain problems formulated in terms of non-topological convergences. Sequential, Fréchet, strongly Fréchet and bisequential topologies are examples of this phenomenon.
    A filter converges to a point of a topological space whenever it contains every open set including this point. Conversely, open sets can be defined with respect to every convergence of filters to the effect that a set is open if it belongs to every filter converging to some of its points; the collection of all open sets for a convergence fulfills all the axioms of open sets of a topology; a convergence is identified with a topology if its convergent filters are precisely those determined by open sets.
    Topologies appeared as an abstraction of convergence of sequences in metric spaces. Aftermath convergence aspects of topologies have been almost abandoned until the investigations of spaces of continuous functions and subsets (of topological spaces) made appeal to the language of convergences of filters, not only of sequences.
    Non-topological convergences appear naturally in analysis, measure theory, optimization and other branches of mathematics: in topological vector spaces there is in general no coarsest topology on the space of continuous linear forms for which the coupling function is continuous; convergence almost everywhere is, in general, non-topological; stability of minimizing set is in general non-topological. It turns out that in such cases a study of non-topological convergences can solve problems formulated in purely topological terms.