Generalization of the notion of convergence that is found in general topology
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. 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
Preliminaries and notation
Denote the power set of a set
by
The upward closure or isotonization in
of a family of subsets
is defined as
![{\displaystyle {\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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7fcb270adfe65998bc77c7f1974b60fddcf4657)
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 ![{\displaystyle F\subseteq C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5772ea654cf37a62403b6bb65f29b71f7a0216)
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
![{\displaystyle \lim {}_{\xi }{\mathcal {F}}:=\left\{x\in X~:~\left(x,{\mathcal {F}}\right)\in \xi \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e37c094d595758b91d5ec68e90c4080bc88dc628)
and if
then define
![{\displaystyle \lim {}_{\xi }^{-1}(x):=\left\{{\mathcal {F}}\subseteq \wp (X)~:~\left(x,{\mathcal {F}}\right)\in \xi \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a789f6caf7d4e63b954ad81defaaebaeccbfa)
so if
then
if and only if
The set
is called the underlying set of
and is denoted by
A preconvergence[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 ![{\displaystyle {\mathcal {G}}\geq {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bba3eee5687b07a56a77b4b16e2daebf9acfc16d)
and if in addition it also has the following property:
- Centered: if
then
- In words, for every
the principal/discrete ultrafilter at
converges to ![{\displaystyle x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a)
then the preconvergence
is called a convergence 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
A preconvergence
can be canonically extended to a relation on
also denoted by
by defining
![{\displaystyle \lim {}_{\xi }{\mathcal {F}}:=\lim {}_{\xi }\left({\mathcal {F}}^{\uparrow X}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52386f3430fb96f8f76cd6023ee35449f3b79d6c)
for all
This extended preconvergence will be isotone on
meaning that if
then
implies
Examples
Convergence induced by a topological space
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 ![{\displaystyle (X,\tau ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b42cadba3d621e09a00a4e133d83ad19f6d548)
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
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
- The standard convergence on the real line
is the convergence
on
defined for all
and all
by:
if and only if ![{\displaystyle {\mathcal {F}}~\geq ~\left\{\left(x-{\frac {1}{n}},x+{\frac {1}{n}}\right)~:~n\in \mathbb {N} \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/deb6896b7c5909c638cdf6de5ae30d960d642219)
- Discrete convergence
- The discrete preconvergence
on a non-empty set
is defined for all
and all
by:
if and only if ![{\displaystyle {\mathcal {F}}~=~\{x\}^{\uparrow X}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02b21fe431c59dcfbe5127f4b5ea13003045b3a1)
- A preconvergence
on
is a convergence if and only if ![{\displaystyle \xi \leq \iota _{X}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e97e5b0853c6d2d02ecf415c8239b783c71931)
- Empty convergence
- The empty preconvergence
on set non-empty
is defined for all
by: ![{\displaystyle \lim {}_{\varnothing _{X}}{\mathcal {F}}:=\emptyset .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9dfe8c69938a27dbbed4299c327c8c009eaacc)
- 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 ![{\displaystyle (X,\tau ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b42cadba3d621e09a00a4e133d83ad19f6d548)
- Chaotic convergence
- The chaotic preconvergence
on set non-empty
is defined for all
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
It is called T1 if
for all
and it is called T0 if
for all distinct
Every T1 preconvergence on a finite set is Hausdorff. Every T1 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.[2]
See also
Citations
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.
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