In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain properties relating elements of X with the family of filters on X. Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence.
Every topological space gives rise to a canonical convergence but there are convergences, known as non-topological convergences, that do not arise from any topological space.
Many topological properties have generalizations to convergence spaces.
Besides its ability to describe notions of convergence that topologies are unable to, the category of convergence spaces has an important categorical property that the category of topological spaces lacks.
The category of topological spaces is not an exponential category (or equivalently, it is not Cartesian closed) although it is contained in the exponential category of pseudotopological spaces, which is itself a subcategory of the (also exponential) category of convergence spaces.
The upward closure or isotonization in
is defined as and similarly the downward closure of
are called equivalent (with respect to subordination
closed under finite intersections, and does not have the empty set as an element (i.e.
A prefilter is any family of sets that is equivalent (with respect to subordination) to some filter or equivalently, it is any family of sets whose upward closure is a filter.
is a prefilter, also called a filter base, if and only if
A filter subbase is any non-empty family of sets with the finite intersection property; equivalently, it is any non-empty family
that is contained as a subset of some filter (or prefilter), in which case the smallest (with respect to
[1] A preconvergence[1][2][4] on a non-empty set
A generalized convergence or a convergence space (respectively a preconvergence space) is a pair consisting of a set
together with a convergence (respectively preconvergence) on
can be canonically extended to a relation on
This extended preconvergence will be isotone on
and elements of this set are called limit points of
The (canonical) convergence associated with or induced by
A (pre)convergence that is induced by some topology on
is called a topological (pre)convergence; otherwise, it is called a non-topological (pre)convergence.
denote the set of continuous maps
that makes the natural coupling
[2] The problem of finding the power has no solution unless
However, if searching for a convergence instead of a topology, then there always exists a convergence that solves this problem (even without local compactness).
[2] In other words, the category of topological spaces is not an exponential category (i.e. or equivalently, it is not Cartesian closed) although it is contained in the exponential category of pseudotopologies, which is itself a subcategory of the (also exponential) category of convergences.
[1] Every T1 preconvergence on a finite set is Hausdorff.
[1] Every T1 convergence on a finite set is discrete.
[1] While the category of topological spaces is not exponential (i.e. Cartesian closed), it can be extended to an exponential category through the use of a subcategory of convergence spaces.