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.

- Definition and notation
- Preliminaries and notation
- Definition of (pre)convergence spaces
- Examples
- Convergence induced by a topological space
- Power
- Other named examples
- Properties
- See also
- Citations
- References

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] }

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

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:

*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:

*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

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*.

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] }

- 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.

A preconvergence on set non-empty is called * Hausdorff * or *T*_{2} if is a singleton set for all ^{ [1] } It is called *T*_{1} if for all and it is called *T*_{0} if for all distinct ^{ [1] } Every *T*_{1} preconvergence on a finite set is Hausdorff.^{ [1] } Every *T*_{1} 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] }

- Cauchy space
- Convergent filter
- Proximity space – A structure describing a notion of "nearness" between subsets.
- Topological space – Mathematical structure with a notion of closeness

In mathematics, a **continuous function** is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be *discontinuous*. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

In the mathematical field of set theory, an **ultrafilter** on a given partially ordered set (poset) *P* is a certain subset of *P,* namely a maximal filter on *P*, that is, a proper filter on *P* that cannot be enlarged to a bigger proper filter on *P*.

In mathematics, **open sets** are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.

In mathematics, the **closure** of a subset *S* of points in a topological space consists of all points in *S* together with all limit points of *S*. The closure of *S* may equivalently be defined as the union of *S* and its boundary, and also as the intersection of all closed sets containing *S*. Intuitively, the closure can be thought of as all the points that are either in *S* or "near" *S*. A point which is in the closure of *S* is a point of closure of *S*. The notion of closure is in many ways dual to the notion of interior.

In geometry, topology, and related branches of mathematics, a **closed set** is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

**Distributions**, also known as **Schwartz distributions** or **generalized functions**, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

In mathematics, a **base** or **basis** for the topology τ of a topological space (*X*, τ) is a family *B* of open subsets of *X* such that every open set is equal to a union of some sub-family of *B*. For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.

In topology and related branches of mathematics, the **Kuratowski closure axioms** are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others.

In mathematics, the **spectrum of a C*-algebra** or **dual of a C*-algebra***A*, denoted *Â*, is the set of unitary equivalence classes of irreducible *-representations of *A*. A *-representation π of *A* on a Hilbert space *H* is **irreducible** if, and only if, there is no closed subspace *K* different from *H* and {0} which is invariant under all operators π(*x*) with *x* ∈ *A*. We implicitly assume that irreducible representation means *non-null* irreducible representation, thus excluding trivial representations on one-dimensional spaces. As explained below, the spectrum *Â* is also naturally a topological space; this is similar to the notion of the spectrum of a ring.

In mathematics, **mixing** is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, *etc*.

In functional analysis and related areas of mathematics, a **sequence space** is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field *K* of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in *K*, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

In general topology and related areas of mathematics, the **final topology** on a set with respect to a family of functions into is the finest topology on that makes those functions continuous.

In topology and related fields of mathematics, a **sequential space** is a topological space that satisfies a very weak axiom of countability.

In topology, a **preclosure operator**, or **Čech closure operator** is a map between subsets of a set, similar to a topological closure operator, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the four Kuratowski closure axioms.

**Uncertainty theory** is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.

In the field of topology, a **Fréchet–Urysohn space** is a topological space with the property that for every subset the closure of in is identical to the *sequential* closure of in Fréchet–Urysohn spaces are a special type of sequential space.

In mathematics, a **polyadic space** is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.

In topology, a subfield of mathematics, *filters* are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called *ultrafilters* have many useful technical properties and they may often be used in place of arbitrary filters.

In mathematics, the **injective tensor product** of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the *completed injective tensor products*. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y with*out* any need to extend definitions from real/complex-valued functions to Y-valued functions.

In mathematics, specifically topology, a **sequence covering map** is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include *sequentially quotient* maps, *sequence coverings*, *1-sequence coverings*, and *2-sequence coverings*. These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness.

- 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.

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