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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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