Cauchy space

Properties and definitions

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.

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 ${\displaystyle A}$ 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 ${\displaystyle \{m:m\ge n\}.}$ 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 ${\displaystyle A\cup \{\mathrm{\infty }\};}$ 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 ${\displaystyle \{1,2,3,\dots \}}$ 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 ${\displaystyle \{1,1/2,1/3,\dots \}.}$

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.