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.
See also: Filters in topology and Ultrafilter.
Denote the power set of a set
X
\wp(X).
X
l{B}\subseteq\wp(X)
l{B}\uparrow:=\left\{S\subseteqX~:~B\subseteqSforsomeB\inl{B}\right\}=cupB
and similarly the of
l{B}
l{B}\downarrow:=\left\{S\subseteqB~:~B\inl{B}\right\}=cupB
l{B}\uparrow=l{B}
l{B}\downarrow=l{B}
l{B}
X.
For any families
l{C}
l{F},
l{C}\leql{F}
C\inl{C},
F\inl{F}
F\subseteqC
l{F}\subseteq\wp(X),
l{C}\leql{F}
l{C}\subseteql{F}\uparrow.
\leq
\wp(\wp(X)).
l{F}\geql{C},
l{C}\leql{F},
l{F}
l{C}
l{C},
l{C}
l{F}.
\geq
l{C}
l{F}
\geq
l{C}\leql{F}
l{F}\leql{C}.
A is a non-empty subset
l{F}\subseteq\wp(X)
X,
\varnothing\not\inl{F}
l{B}
\varnothing\not\inl{B} ≠ \varnothing
B,C\inl{B},
A\inl{B}
A\subseteqB\capC.
l{B}
\subseteq
\leq
l{B}
X
\operatorname{Filters}(X)
\operatorname{Prefilters}(X),
\operatorname{FilterSubbases}(X),
\operatorname{UltraFilters}(X)
X
x\inX
\{x\}\uparrow.
For any
\xi\subseteqX x \wp(\wp(X)),
l{F}\subseteq\wp(X)
\lim{}\xil{F}:=\left\{x\inX~:~\left(x,l{F}\right)\in\xi\right\}
and if
x\inX
\lim
-1 | |
{} | |
\xi |
(x):=\left\{l{F}\subseteq\wp(X)~:~\left(x,l{F}\right)\in\xi\right\}
so if
\left(x,l{F}\right)\inX x \wp(\wp(X))
x\in\lim{}\xil{F}
\left(x,l{F}\right)\in\xi.
X
\xi
\left|\xi\right|:=X.
A on a non-empty set
X
\xi\subseteqX x \operatorname{Filters}(X)
l{F},l{G}\in\operatorname{Filters}(X)
l{F}\leql{G}
\lim{}\xil{F}\subseteq\lim{}\xil{G}
l{F}
l{G}\geql{F}.
and if in addition it also has the following property:
x\inX
x\in\lim{}\xi\left(\{x\}\uparrow\right)
x\inX,
x
x.
then the preconvergence
\xi
X.
X
X.
A preconvergence
\xi\subseteqX x \operatorname{Filters}(X)
X x \operatorname{Prefilters}(X),
\xi,
\lim{}\xil{F}:=\lim{}\xi\left(l{F}\uparrow\right)
for all
l{F}\in\operatorname{Prefilters}(X).
\operatorname{Prefilters}(X),
l{F},l{G}\in\operatorname{Prefilters}(X)
l{F}\leql{G}
\lim{}\xil{F}\subseteq\lim{}\xil{G}.
Let
(X,\tau)
X ≠ \varnothing.
l{F}\in\operatorname{Filters}(X)
l{F}
x\inX
(X,\tau),
l{F}\tox
(X,\tau),
l{F}\geql{N}(x),
l{N}(x)
x
(X,\tau).
x\inX
l{F}\tox
(X,\tau)
\lim{}(X,l{F},
\lim{}Xl{F},
\liml{F},
l{F}
(X,\tau).
(X,\tau)
X,
\xi\tau,
x\inX
l{F}\in\operatorname{Filters}(X)
x\in\lim
{} | |
\xi\tau |
l{F}
l{F}\tox
(X,\tau).
\lim
{} | |
\xi\tau |
l{F}:=\lim{}(X,l{F}
l{F}\in\operatorname{Filters}(X).
A (pre)convergence that is induced by some topology on
X
Let
(X,\tau)
(Z,\sigma)
C:=C\left((X,\tau);(Z,\sigma)\right)
f:(X,\tau)\to(Z,\sigma).
\theta
C
\left\langlex,f\right\rangle=f(x)
(X,\tau) x \left(C,\theta\right)\to(Z,\sigma).
(X,\tau)
R
\nu
X
x\inX=R
l{F}\in\operatorname{Filters}(X)
x\in\lim{}\nul{F}
l{F}~\geq~\left\{\left(x-
1{n}, | |
x |
+
1{n} | |
\right) |
~:~n\inN\right\}.
\iotaX
X
x\inX
l{F}\in\operatorname{Filters}(X)
x\in\lim
{} | |
\iotaX |
l{F}
l{F}~=~\{x\}\uparrow.
A preconvergence
\xi
X
\xi\leq\iotaX.
\varnothingX
X
l{F}\in\operatorname{Filters}(X)
\lim
{} | |
\varnothingX |
l{F}:=\emptyset.
Although it is a preconvergence on
X,
X.
X ≠ \varnothing
\tau
X,
x\inX
x
(X,\tau).
oX
X
l{F}\in\operatorname{Filters}(X)
\lim
{} | |
oX |
l{F}:=X.
X
X
X
A preconvergence
\xi
X
\lim{}\xil{F}
l{F}\in\operatorname{Filters}(X).
\lim{}\xi\left(\{x\}\uparrow\right)\subseteq\{x\}
x\inX
\operatorname{lim}-1{}\xi(x) ≠ \operatorname{lim}-1{}\xi(y)
x,y\inX.
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.