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Convergence Theory
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.
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