Convergence space

In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a 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 , that do not arise from any topological space. Examples of convergences that are in general non-topological include convergence in measure and almost everywhere convergence. 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.
Definition and notation.
Preliminaries and notation.
Denote the power set of a set formula_1 by formula_2 The or in formula_1 of a family of subsets formula_4 is defined as
and similarly the of formula_6 is formula_7 
If formula_8 (resp. formula_9) then formula_6 is said to be (resp. ) in formula_11 
For any families formula_12 and formula_13 declare that 
or equivalently, if formula_18 then formula_14 if and only if formula_20 The relation formula_21 defines a preorder on formula_22 If formula_23 which by definition means formula_24 then formula_25 is said to be formula_12 and also formula_27 and formula_12 is said to be formula_29 The relation formula_30 is called . Two families formula_12 and formula_25 are called ( formula_30) if formula_14 and formula_35 
A is a non-empty subset formula_36 that is upward closed in formula_37 closed under finite intersections, and does not have the empty set as an element (i.e. formula_38). A is any family of sets that is equivalent (with respect to subordination) to filter or equivalently, it is any family of sets whose upward closure is a filter. A family formula_6 is a prefilter, also called a , if and only if formula_40 and for any formula_41 there exists some formula_42 such that formula_43 
A is any non-empty family of sets with the finite intersection property; equivalently, it is any non-empty family formula_6 that is contained as a subset of some filter (or prefilter), in which case the smallest (with respect to formula_45 or formula_46) filter containing formula_6 is called () . 
The set of all filters (resp. prefilters, filter subbases, ultrafilters) on formula_1 will be denoted by formula_49 (resp. formula_50 formula_51 formula_52). 
The or filter on formula_1 at a point formula_54 is the filter formula_55 
Definition of (pre)convergence spaces.
For any formula_56 if formula_36 then define
and if formula_54 then define
so if formula_61 then formula_62 if and only if formula_63 The set formula_1 is called the of formula_65 and is denoted by formula_66 
A on a non-empty set formula_1 is a binary relation formula_68 with the following property:
and if in addition it also has the following property: 
then the preconvergence formula_65 is called a on formula_11 
A or a (resp. a ) is a pair consisting of a set formula_1 together with a convergence (resp. preconvergence) on formula_11 
A preconvergence formula_68 can be canonically extended to a relation on formula_74 also denoted by formula_75 by defining 
for all formula_77 This extended preconvergence will be isotone on formula_50 meaning that if formula_79 then formula_80 implies formula_81
Examples.
Convergence induced by a topological space.
Let formula_82 be a topological space with formula_83 If formula_84 then formula_25 is said to to a point formula_54 in formula_87 written formula_88 in formula_87 if formula_90 where formula_91 denotes the neighborhood filter of formula_92 in formula_93 The set of all formula_54 such that formula_88 in formula_82 is denoted by formula_97 formula_98 or simply formula_99 and elements of this set are called of formula_25 in formula_93 
The () or formula_82 is the convergence on formula_37 denoted by formula_104 defined for all formula_54 and all formula_84 by:
Equivalently, it is defined by formula_110 for all formula_111 
A (pre)convergence that is induced by some topology on formula_1 is called a ; otherwise, it is called a . 
Power.
Let formula_82 and formula_114 be topological spaces and let formula_115 denote the set of continuous maps formula_116 The is the coarsest topology formula_117 on formula_118 that makes the natural coupling formula_119 into a continuous map formula_120 
The problem of finding the power has no solution unless formula_82 is locally compact. However, if searching for a convergence instead of a topology, then there always exists a convergence that solves this problem (even without local compactness). 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.
Properties.
A preconvergence formula_65 on set non-empty formula_1 is called or if formula_158 is a singleton set for all formula_111 It is called if formula_160 for all formula_54 and it is called if formula_162 for all distinct formula_163 
Every preconvergence on a finite set is Hausdorff. Every convergence on a finite set is discrete. 
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.