Ingeneral topology andanalysis, aCauchy space is a generalization ofmetric spaces anduniform spaces 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 aCauchy filter, in order to studycompleteness intopological spaces. Thecategory of Cauchy spaces andCauchy continuous maps isCartesian closed, and contains the category ofproximity spaces.

Let${\displaystyle X}$ be a set,${\displaystyle \mathcal{P}(X)}$ thepower set of${\displaystyle X}$, and assume allfilters areproper (that is, a filter may not contain the empty set).

A Cauchy space is a pair${\displaystyle (X,C)}$ consisting of a set${\displaystyle X}$ together with afamily${\displaystyle C\subseteq \mathcal{P}(\mathcal{P}(X))}$ of (proper) filters on${\displaystyle X}$ having all of the following properties:

1. For each${\displaystyle x\in X,}$ the discreteultrafilter at${\displaystyle x,}$ denoted by${\displaystyle U(x),}$ is in${\displaystyle C.}$
2. If${\displaystyle F\in C,}$${\displaystyle G}$ is a proper filter, and${\displaystyle F}$ is a subset of${\displaystyle G,}$ then${\displaystyle G\in C.}$
3. If${\displaystyle F,G\in C}$ and if each member of${\displaystyle F}$ intersects each member of${\displaystyle G,}$ then${\displaystyle F\cap G\in C.}$

An element of${\displaystyle C}$ is called aCauchy filter, and a map${\displaystyle f}$ between Cauchy spaces${\displaystyle (X,C)}$ and${\displaystyle (Y,D)}$ isCauchy continuous if${\displaystyle \uparrow f(C)\subseteq D}$; that is, the image of each Cauchy filter in${\displaystyle X}$ is a Cauchy filter base in${\displaystyle Y.}$

Any Cauchy space is also aconvergence space, where a filter${\displaystyle F}$ converges to${\displaystyle x}$ if${\displaystyle F\cap U(x)}$ is Cauchy. In particular, a Cauchy space carries a naturaltopology.

• Anyuniform space (hence anymetric space,topological vector space, ortopological group) is a Cauchy space; seeCauchy filter for definitions.
• Alattice-ordered group carries a natural Cauchy structure.
• Anydirected set${\displaystyle A}$ may be made into a Cauchy space by declaring a filter${\displaystyle F}$ to be Cauchy if,given any element${\displaystyle n\in A,}$there is an element${\displaystyle U\in F}$ such that${\displaystyle U}$ is either asingleton or asubset of the tail${\displaystyle \{m:m\ge n\}.}$ Then given any other Cauchy space${\displaystyle X,}$ theCauchy-continuous functions from${\displaystyle A}$ to${\displaystyle X}$ are the same as theCauchy nets in${\displaystyle X}$ indexed by${\displaystyle A.}$ If${\displaystyle X}$ iscomplete, then such a function may be extended to the completion of${\displaystyle A,}$ which may be written${\displaystyle A\cup \{\mathrm{\infty }\};}$ the value of the extension at${\displaystyle \mathrm{\infty }}$ will be the limit of the net. In the case where${\displaystyle A}$ is the set${\displaystyle \{1,2,3,\dots \}}$ ofnatural numbers (so that a Cauchy net indexed by${\displaystyle A}$ is the same as aCauchy sequence), then${\displaystyle A}$ receives the same Cauchy structure as the metric space${\displaystyle \{1,1/2,1/3,\dots \}.}$

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

• Characterizations of the category of topological spaces – Multiple equivalent ways to define a topological spacePages displaying short descriptions of redirect targets
• Convergence space – Generalization of the notion of convergence that is found in general topology
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results
• Pretopological space – Generalized topological space
• Proximity space – Structure describing a notion of "nearness" between subsets

• Eva Lowen-Colebunders (1989).Function Classes of Cauchy Continuous Maps. Dekker, New York, 1989.
• Schechter, Eric (1996).Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press.ISBN 978-0-12-622760-4.OCLC 175294365.

• General topology

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