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  by   The  or  in  of a family of subsets  is defined as

and similarly the  of  is  
If  (resp. ) then  is said to be  (resp. ) 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 (resp. prefilters, filter subbases, ultrafilters) on  will be denoted by  (resp.   ). 
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:

: if  then  implies 
 In words, any limit point of  is necessarily a limit point of any finer/subordinate family 

and if in addition it also has the following property:

: if  then 
 In words, for every  the principal/discrete ultrafilter at  converges to 

then the preconvergence  is called a  on   
A  or a  (resp. a ) is a pair consisting of a set  together with a convergence (resp. 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 ℝ The  is the convergence  on  defined for all  and all  by: 
 if and only if 

Discrete convergence The   on set non-empty  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  is a singleton set for all   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.

See also

Citations

References 

  
 
 
  

 
Mathematical structures