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 notation

See also: Filters in topology and Ultrafilter

Preliminaries and notation

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 spaces

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

Examples

Convergence induced by a topological space

See also: Convergent filter

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.

Power

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 examples

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 a non-empty set 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.

Properties

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 also

Citations

  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

References

  • 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.