Cauchy Space - Examples

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 A may be made into a Cauchy space by declaring a filter F to be Cauchy if, given any element n of A, there is an element U of F such that U is either a singleton or a subset of the tail {m | m ≥ n}. Then given any other Cauchy space X, the Cauchy-continuous functions from A to X are the same as the Cauchy nets in X indexed by A. If X is complete, then such a function may be extended to the completion of A, which may be written A ∪ {∞}; the value of the extension at ∞ will be the limit of the net. In the case where A is the set {1, 2, 3, …} of natural numbers (so that a Cauchy net indexed by A is the same as a Cauchy sequence), then A receives the same Cauchy structure as the metric space {1, 1/2, 1/3, …}.