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. 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.

Preliminaries and notation
Denote the power set of a set $$X$$ by $$\wp(X).$$ The  or  in $$X$$ of a family of subsets $$\mathcal{B} \subseteq \wp(X)$$ is defined as


 * $$\mathcal{B}^{\uparrow X} := \left\{ S \subseteq X ~:~ B \subseteq S \text{ for some } B \in \mathcal{B} \, \right\} = \bigcup_{B \in \mathcal{B}} \left\{ S ~:~ B \subseteq S \subseteq X \right\}$$

and similarly the of $$\mathcal{B}$$ is $$\mathcal{B}^{\downarrow} := \left\{ S \subseteq B ~:~ B \in \mathcal{B} \, \right\} = \bigcup_{B \in \mathcal{B}} \wp(B).$$ If $$\mathcal{B}^{\uparrow X} = \mathcal{B}$$ (resp. $$\mathcal{B}^{\downarrow} = \mathcal{B}$$) then $$\mathcal{B}$$ is said to be (resp. ) in $$X.$$

For any families $$\mathcal{C}$$ and $$\mathcal{F},$$ declare that
 * $$\mathcal{C} \leq \mathcal{F}$$ if and only if for every $$C \in \mathcal{C},$$ there exists some $$F \in \mathcal{F}$$ such that $$F \subseteq C$$

or equivalently, if $$\mathcal{F} \subseteq \wp(X),$$ then $$\mathcal{C} \leq \mathcal{F}$$ if and only if $$\mathcal{C} \subseteq \mathcal{F}^{\uparrow X}.$$ The relation $$\,\leq\,$$ defines a preorder on $$\wp(\wp(X)).$$ If $$\mathcal{F} \geq \mathcal{C},$$ which by definition means $$\mathcal{C} \leq \mathcal{F},$$ then $$\mathcal{F}$$ is said to be $$\mathcal{C}$$ and also  $$\mathcal{C},$$ and $$\mathcal{C}$$ is said to be  $$\mathcal{F}.$$ The relation $$\,\geq\,$$ is called. Two families $$\mathcal{C}$$ and $$\mathcal{F}$$ are called ( $$\,\geq\,$$) if $$\mathcal{C} \leq \mathcal{F}$$ and $$\mathcal{F} \leq \mathcal{C}.$$

A is a non-empty subset $$\mathcal{F} \subseteq \wp(X)$$ that is upward closed in $$X,$$ closed under finite intersections, and does not have the empty set as an element (i.e. $$\varnothing \not\in \mathcal{F}$$). 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 $$\mathcal{B}$$ is a prefilter, also called a, if and only if $$\varnothing \not\in \mathcal{B} \neq \varnothing$$ and for any $$B, C \in \mathcal{B},$$ there exists some $$A \in \mathcal{B}$$ such that $$A \subseteq B \cap C.$$ A is any non-empty family of sets with the finite intersection property; equivalently, it is any non-empty family $$\mathcal{B}$$ that is contained as a subset of some filter (or prefilter), in which case the smallest (with respect to $$\subseteq$$ or $$\leq$$) filter containing $$\mathcal{B}$$ is called. The set of all filters (resp. prefilters, filter subbases, ultrafilters) on $$X$$ will be denoted by $$\operatorname{Filters}(X)$$ (resp. $$\operatorname{Prefilters}(X),$$ $$\operatorname{FilterSubbases}(X),$$ $$\operatorname{UltraFilters}(X)$$). The or  filter on $$X$$ at a point $$x \in X$$ is the filter $$\{ x \}^{\uparrow X}.$$

Definition of (pre)convergence spaces
For any $$\xi \subseteq X \times \wp(\wp(X)),$$ if $$\mathcal{F} \subseteq \wp(X)$$ then define


 * $$\lim {}_\xi \mathcal{F} := \left\{ x \in X ~:~ \left( x, \mathcal{F} \right) \in \xi \right\}$$

and if $$x \in X$$ then define


 * $$\lim {}^{-1}_{\xi} (x) := \left\{ \mathcal{F} \subseteq \wp(X) ~:~ \left( x, \mathcal{F} \right) \in \xi \right\}$$

so if $$\left( x, \mathcal{F} \right) \in X \times \wp(\wp(X))$$ then $$x \in \lim {}_{\xi} \mathcal{F}$$ if and only if $$\left( x, \mathcal{F} \right) \in \xi.$$ The set $$X$$ is called the  of $$\xi$$ and is denoted by $$\left| \xi \right| := X.$$

A  on a non-empty set $$X$$ is a binary relation $$\xi \subseteq X \times \operatorname{Filters}(X)$$ with the following property:

 : if $$\mathcal{F}, \mathcal{G} \in \operatorname{Filters}(X)$$ then $$\mathcal{F} \leq \mathcal{G}$$ implies $$\lim {}_{\xi} \mathcal{F} \subseteq \lim {}_{\xi} \mathcal{G}$$ 
 * In words, any limit point of $$\mathcal{F}$$ is necessarily a limit point of any finer/subordinate family $$\mathcal{G} \geq \mathcal{F}.$$

and if in addition it also has the following property:

: if $$x \in X$$ then $$x \in \lim {}_{\xi} \left( \{ x \}^{\uparrow X} \right)$$ 
 * In words, for every $$x \in X,$$ the principal/discrete ultrafilter at $$x$$ converges to $$x.$$

then the preconvergence $$\xi$$ is called a on $$X.$$ A or a  (resp. a ) is a pair consisting of a set $$X$$ together with a convergence (resp. preconvergence) on $$X.$$

A preconvergence $$\xi \subseteq X \times \operatorname{Filters}(X)$$ can be canonically extended to a relation on $$X \times \operatorname{Prefilters}(X),$$ also denoted by $$\xi,$$ by defining


 * $$\lim {}_{\xi} \mathcal{F} := \lim {}_{\xi} \left( \mathcal{F}^{\uparrow X} \right)$$

for all $$\mathcal{F} \in \operatorname{Prefilters}(X).$$ This extended preconvergence will be isotone on $$\operatorname{Prefilters}(X),$$ meaning that if $$\mathcal{F}, \mathcal{G} \in \operatorname{Prefilters}(X)$$ then $$\mathcal{F} \leq \mathcal{G}$$ implies $$\lim {}_{\xi} \mathcal{F} \subseteq \lim {}_{\xi} \mathcal{G}.$$

Convergence induced by a topological space
Let $$(X, \tau)$$ be a topological space with $$X \neq \varnothing.$$ If $$\mathcal{F} \in \operatorname{Filters}(X)$$ then $$\mathcal{F}$$ is said to  to a point $$x \in X$$ in $$(X, \tau),$$ written $$\mathcal{F} \to x$$ in $$(X, \tau),$$ if $$\mathcal{F} \geq \mathcal{N}(x),$$ where $$\mathcal{N}(x)$$ denotes the neighborhood filter of $$x$$ in $$(X, \tau).$$  The set of all $$x \in X$$ such that $$\mathcal{F} \to x$$ in $$(X, \tau)$$ is denoted by $$\lim {}_{(X, \tau)} \mathcal{F},$$ $$\lim {}_X \mathcal{F},$$ or simply $$\lim \mathcal{F},$$ and elements of this set are called  of $$\mathcal{F}$$ in $$(X, \tau).$$ The  or  $$(X, \tau)$$ is the convergence on $$X,$$ denoted by $$\xi_{\tau},$$ defined for all $$x \in X$$ and all $$\mathcal{F} \in \operatorname{Filters}(X)$$ by:
 * $$x \in \lim {}_{\xi_{\tau}} \mathcal{F}$$ if and only if $$\mathcal{F} \to x$$ in $$(X, \tau).$$

Equivalently, it is defined by $$\lim {}_{\xi_{\tau}} \mathcal{F} := \lim {}_{(X, \tau)} \mathcal{F}$$ for all $$\mathcal{F} \in \operatorname{Filters}(X).$$

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

Power
Let $$(X, \tau)$$ and $$(Z, \sigma)$$ be topological spaces and let $$C := C\left( (X, \tau); (Z, \sigma) \right)$$ denote the set of continuous maps $$f : (X, \tau) \to (Z, \sigma).$$ The  is the coarsest topology $$\theta$$ on $$C$$ that makes the natural coupling $$\left\langle x, f \right\rangle = f(x)$$ into a continuous map $$(X, \tau) \times \left( C, \theta \right) \to (Z, \sigma).$$ The problem of finding the power has no solution unless $$(X, \tau)$$ 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 $$\mathbb{R}$$: The is the convergence $$\nu$$ on $$X$$ defined for all $$x \in X = \mathbb{R}$$ and all $$\mathcal{F} \in \operatorname{Filters}(X)$$ by:
 * $$x \in \lim {}_{\nu} \mathcal{F}$$ if and only if $$\mathcal{F} ~\geq~ \left\{ \left( x - \frac1{n}, x + \frac1{n} \right) ~:~ n \in \mathbb{N} \right\}.$$


 * Discrete convergence: The $$\iota_{X}$$ on a non-empty set $$X$$ is defined for all $$x \in X$$ and all $$\mathcal{F} \in \operatorname{Filters}(X)$$ by:
 * $$x \in \lim {}_{\iota_{X}} \mathcal{F}$$ if and only if $$\mathcal{F} ~=~ \{ x \}^{\uparrow X}.$$
 * A preconvergence $$\xi$$ on $$X$$ is a convergence if and only if $$\xi \leq \iota_{X}.$$


 * Empty convergence: The $$\varnothing_{X}$$ on set non-empty $$X$$ is defined for all $$\mathcal{F} \in \operatorname{Filters}(X)$$ by: $$\lim {}_{\varnothing_{X}} \mathcal{F} := \emptyset.$$


 * Although it is a preconvergence on $$X,$$ it is a convergence on $$X.$$ The empty preconvergence on $$X \neq \varnothing$$ is a non-topological preconvergence because for every topology $$\tau$$ on $$X,$$ the neighborhood filter at any given point $$x \in X$$ necessarily converges to $$x$$ in $$(X, \tau).$$


 * Chaotic convergence: The $$o_{X}$$ on set non-empty $$X$$ is defined for all $$\mathcal{F} \in \operatorname{Filters}(X)$$ by: $$\lim {}_{o_{X}} \mathcal{F} := X.$$ The chaotic preconvergence on $$X$$ is equal to the canonical convergence induced by $$X$$ when $$X$$ is endowed with the indiscrete topology.

Properties
A preconvergence $$\xi$$ on set non-empty $$X$$ is called or $T_{2}$ if $$\lim {}_{\xi} \mathcal{F}$$ is a singleton set for all $$\mathcal{F} \in \operatorname{Filters}(X).$$  It is called $T_{1}$ if $$\lim {}_{\xi} \left( \{ x \}^{\uparrow X} \right) \subseteq \{ x \}$$ for all $$x \in X$$ and it is called $T_{0}$ if $$\operatorname{lim}^{-1}{}_{\xi} (x) \neq \operatorname{lim}^{-1}{}_{\xi} (y)$$ for all distinct $$x, y \in X.$$ Every $T_{1}$ preconvergence on a finite set is Hausdorff. Every $T_{1}$ 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.