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. An example of convergence that is in general non-topological is 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 by The or in of a family of subsets is defined as

and similarly the of is If (respectively ) then is said to be (respectively ) in

For any families and declare that

if and only if for every there exists some such that

or equivalently, if then if and only if The relation defines a preorder on If which by definition means then is said to be and also and is said to be The relation is called . Two families and are called ( ) if and

A is a non-empty subset that is upward closed in closed under finite intersections, and does not have the empty set as an element (i.e. ). 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 is a prefilter, also called a , if and only if and for any there exists some such that A 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 or ) filter containing is called () . The set of all filters (respectively prefilters, filter subbases, ultrafilters) on will be denoted by (respectively ). The or filter on at a point is the filter

Definition of (pre)convergence spaces

For any if then define

and if then define

so if then if and only if The set is called the of and is denoted by

A on a non-empty set is a binary relation with the following property:

<ol> <li>: if then implies

In words, any limit point of is necessarily a limit point of any finer/subordinate family </li>

</ol>

and if in addition it also has the following property:

<ol start=2> <li>: if then

In words, for every the principal/discrete ultrafilter at converges to </li>

</ol>

then the preconvergence is called a on A or a (respectively a ) is a pair consisting of a set together with a convergence (respectively preconvergence) on

A preconvergence can be canonically extended to a relation on also denoted by by defining

for all This extended preconvergence will be isotone on meaning that if then implies

Examples

Convergence induced by a topological space

Let be a topological space with If then is said to to a point in written in if where denotes the neighborhood filter of in The set of all such that in is denoted by or simply and elements of this set are called of in The () or is the convergence on denoted by defined for all and all by:

if and only if in

Equivalently, it is defined by for all

A (pre)convergence that is induced by some topology on is called a ; otherwise, it is called a .

Power

Let and be topological spaces and let denote the set of continuous maps The is the coarsest topology on that makes the natural coupling into a continuous map The problem of finding the power has no solution unless 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.

Other named examples

Standard convergence on : The is the convergence on defined for all and all by:

if and only if

Discrete convergence: The on a non-empty set is defined for all and all by:

if and only if

A preconvergence on is a convergence if and only if

Empty convergence: The on set non-empty is defined for all by:

Although it is a preconvergence on it is a convergence on The empty preconvergence on is a non-topological preconvergence because for every topology on the neighborhood filter at any given point necessarily converges to in

Chaotic convergence: The on set non-empty is defined for all by: The chaotic preconvergence on is equal to the canonical convergence induced by when is endowed with the indiscrete topology.

Properties

A preconvergence on set non-empty is called or if, for all , is either a singleton set or empty. It is called if for all and it is called if for all distinct 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.