Cauchy space

Throughout, ${\displaystyle X}$ is a set, ${\displaystyle \wp (X)}$ denotes the power set of ${\displaystyle X,}$ and all filters are assumed to be proper/non-degenerate (i.e. a filter may not contain the empty set).

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

1. For each ${\displaystyle x\in X,}$ the discrete ultrafilter 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 a Cauchy filter, and a map ${\displaystyle f}$ between Cauchy spaces ${\displaystyle (X,C)}$ and ${\displaystyle (Y,D)}$ is Cauchy 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 a convergence 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 natural topology.

• 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 ${\displaystyle A}$ 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 a singleton or a subset of the tail ${\displaystyle \{m:m\geq n\}.}$ Then given any other Cauchy space ${\displaystyle X,}$ the Cauchy-continuous functions from ${\displaystyle A}$ to ${\displaystyle X}$ are the same as the Cauchy nets in ${\displaystyle X}$ indexed by ${\displaystyle A.}$ If ${\displaystyle X}$ is complete, then such a function may be extended to the completion of ${\displaystyle A,}$ which may be written ${\displaystyle A\cup \{\infty \};}$ the value of the extension at ${\displaystyle \infty }$ will be the limit of the net. In the case where ${\displaystyle A}$ is the set ${\displaystyle \{1,2,3,\ldots \}}$ of natural numbers (so that a Cauchy net indexed by ${\displaystyle A}$ is the same as a Cauchy sequence), then ${\displaystyle A}$ receives the same Cauchy structure as the metric space ${\displaystyle \{1,1/2,1/3,\ldots \}.}$

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.

• Characterizations of the category of topological spaces
• 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.