Editing Cauchy space

{{Short description|Concept in general topology and analysis}}
In [[general topology]] and [[mathematical analysis|analysis]], a '''Cauchy space''' is a generalization of [[metric space]]s and [[uniform space]]s for which the notion of Cauchy convergence still makes sense.  Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a [[Cauchy filter]], in order to study [[Complete metric space|completeness]] in [[topological space]]s. The [[Category (mathematics)|category]] of Cauchy spaces and ''Cauchy continuous maps'' is [[Cartesian closed]], and contains the category of [[proximity space]]s.

==Definition==

Throughout, <math>X</math> is a set, <math>\wp(X)</math> denotes the [[power set]] of <math>X,</math> and all [[Filter (set theory)|filters]] are assumed to be [[Proper filter|proper/non-degenerate]] (i.e. a filter may not contain the empty set). 

A Cauchy space is a pair <math>(X, C)</math> consisting of a set <math>X</math> together with a [[Family of sets|family]] <math>C \subseteq \wp(\wp(X))</math> of (proper) filters on <math>X</math> having all of the following properties:
# For each <math>x \in X,</math> the discrete [[Ultrafilter (set theory)|ultrafilter]] at <math>x,</math> denoted by <math>U(x),</math> is in <math>C.</math>
# If <math>F \in C,</math> <math>G</math> is a proper filter, and <math>F</math> is a subset of <math>G,</math> then <math>G \in C.</math> 
# If <math>F, G \in C</math> and if each member of <math>F</math> intersects each member of <math>G,</math> then <math>F \cap G \in C.</math> 
An element of <math>C</math> is called a '''Cauchy filter''', and a map <math>f</math> between Cauchy spaces <math>(X, C)</math> and <math>(Y, D)</math> is '''Cauchy continuous''' if <math>\uparrow f(C) \subseteq D</math>; that is, the image of each Cauchy filter in <math>X</math> is a Cauchy filter base in <math>Y.</math>

==Properties and definitions==

Any Cauchy space is also a [[convergence space]], where a filter <math>F</math> converges to <math>x</math> if <math>F \cap U(x)</math> is Cauchy. In particular, a Cauchy space carries a natural [[Topology (structure)|topology]].

==Examples==

* Any [[uniform space]] (hence any [[metric space]], [[topological vector space]], or [[topological group]]) is a Cauchy space; see [[Cauchy filter]] for definitions.
* A [[lattice-ordered group]] carries a natural Cauchy structure.
* Any [[directed set]] <math>A</math> may be made into a Cauchy space by declaring a filter <math>F</math> to be Cauchy if, [[given any]] element <math>n \in A,</math> [[there is]] an element <math>U \in F</math> such that <math>U</math> is either a [[singleton (set theory)|singleton]] or a [[subset]] of the tail <math>\{m : m \geq n\}.</math> Then given any other Cauchy space <math>X,</math> the [[Cauchy-continuous function]]s from <math>A</math> to <math>X</math> are the same as the [[Cauchy net]]s in <math>X</math> indexed by <math>A.</math> If <math>X</math> is [[complete space|complete]], then such a function may be extended to the completion of <math>A,</math> which may be written <math>A \cup \{\infty\};</math> the value of the extension at <math>\infty</math> will be the limit of the net. In the case where <math>A</math> is the set <math>\{1, 2, 3, \ldots\}</math> of [[natural number]]s (so that a Cauchy net indexed by <math>A</math> is the same as a [[Cauchy sequence]]), then <math>A</math> receives the same Cauchy structure as the metric space <math>\{1, 1/2, 1/3, \ldots\}.</math> 

==Category of Cauchy spaces==

The natural notion of [[morphism]] between Cauchy spaces is that of a [[Cauchy-continuous function]], a concept that had earlier been studied for uniform spaces.

==See also==

* {{annotated link|Characterizations of the category of topological spaces}}
* {{annotated link|Convergence space}}
* {{annotated link|Filters in topology}}
* {{annotated link|Pretopological space}}
* {{annotated link|Proximity space}}

==References==

{{reflist|group=note}}
{{reflist}}

* Eva Lowen-Colebunders (1989). <cite>Function Classes of Cauchy Continuous Maps</cite>. Dekker, New York, 1989.
* {{Schechter Handbook of Analysis and Its Foundations}} <!-- {{sfn|Schechter|1996|p=}} -->

{{Topology}}

[[Category:General topology]]