Cauchy-continuous Function - Generalisations

Generalisations

Cauchy continuity makes sense in situations more general than metric spaces, but then one must move from sequences to nets (or equivalently filters). The definition above applies, as long as the Cauchy sequence (x₁, x₂, …) is replaced with an arbitrary Cauchy net. Equivalently, a function f is Cauchy-continuous if and only if, given any Cauchy filter F on X, then f(F) is a Cauchy filter on Y. This definition agrees with the above on metric spaces, but it also works for uniform spaces and, most generally, for Cauchy spaces.

Any directed set A may be made into a Cauchy space. Then given any space Y, the Cauchy nets in Y indexed by A are the same as the Cauchy-continuous functions from A to Y. If Y is complete, then the extension of the function to A ∪ {∞} will give the value of the limit of the net. (This generalises the example of sequences above, where 0 is to be interpreted as 1/∞.)