Convergence space

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.[1] 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.[2]

Definition and notationEdit

Preliminaries and notationEdit

Denote the power set of a set   by   The upward closure or isotonization in  [3] of a family of subsets   is defined as

 

and similarly the downward closure of   is   If   (resp.  ) then   is said to be upward closed (resp. downward closed) 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 subordinate to   and also finer than   and   is said to be coarser than   The relation   is called subordination. Two families   and   are called equivalent (with respect to subordination  ) if   and  

A filter on a set   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 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. A family   is a prefilter, also called a filter base, if and only if   and for any   there exists some   such that   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   or  ) filter containing   is called the filter (on  ) generated by  . The set of all filters (resp. prefilters, filter subbases, ultrafilters) on   will be denoted by   (resp.      ). The principal or discrete filter on   at a point   is the filter  

Definition of (pre)convergence spacesEdit

For any   if   then define

 

and if   then define

 

so if   then   if and only if   The set   is called the underlying set of   and is denoted by  [1]

A preconvergence[1][2][4] on a non-empty set   is a binary relation   with the following property:

  1. Isotone: 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:

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

then the preconvergence   is called a convergence[1] on   A generalized convergence or a convergence space (resp. a preconvergence space) is a pair consisting of a set   together with a convergence (resp. preconvergence) on  [1]

A preconvergence   can be canonically extended to a relation on   also denoted by   by defining[1]

 

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

ExamplesEdit

Convergence induced by a topological spaceEdit

Let   be a topological space with   If   then   is said to converge 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 limit points of   in   The (canonical) convergence associated with or induced by   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 topological (pre)convergence; otherwise, it is called a non-topological (pre)convergence.

PowerEdit

Let   and   be topological spaces and let   denote the set of continuous maps   The power with respect to   and   is the coarsest topology   on   that makes the natural coupling   into a continuous map  [2] 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).[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.[2]

Other named examplesEdit

Standard convergence on ℝ
The standard convergence on the real line   is the convergence   on   defined for all   and all  [1] by:
  if and only if  
Discrete convergence
The discrete preconvergence   on set non-empty   is defined for all   and all  [1] by:
  if and only if  
A preconvergence   on   is a convergence if and only if  [1]
Empty convergence
The empty preconvergence   on set non-empty   is defined for all  [1] by:  
Although it is a preconvergence on   it is not 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 chaotic preconvergence   on set non-empty   is defined for all  [1] by:   The chaotic preconvergence on   is equal to the canonical convergence induced by   when   is endowed with the indiscrete topology.

PropertiesEdit

A preconvergence   on set non-empty   is called Hausdorff or T2 if   is a singleton set for all  [1] It is called T1 if   for all   and it is called T0 if   for all distinct  [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.[2]

See alsoEdit

CitationsEdit

  1. ^ a b c d e f g h i j k l m n o Dolecki & Mynard 2016, pp. 55–77.
  2. ^ a b c d e f Dolecki 2009, pp. 1–51
  3. ^ Dolecki & Mynard 2016, pp. 27–29.
  4. ^ Dolecki & Mynard 2014, pp. 1–25

ReferencesEdit

  • Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
  • Dolecki, Szymon (2009). Mynard, Frédéric; Pearl, Elliott (eds.). "An initiation into convergence theory" (PDF). Beyond Topology. Contemporary Mathematics Series A.M.S. 486: 115–162. Retrieved 14 January 2021.
  • Dolecki, Szymon; Mynard, Frédéric (2014). "A unified theory of function spaces and hyperspaces: local properties" (PDF). Houston J. Math. 40 (1): 285–318. Retrieved 14 January 2021.
  • Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.