called a ; explicitly, a it is a uniformly continuous bijection whose inverse is also uniformly continuous. A is an injective uniformly continuous map between uniform spaces whose inverse is also uniformly continuous, where the image has the subspace uniformity inherited from Completeness Generalizing the notion of complete metric space, one can also define completeness for uniform spaces. Instead of working with Cauchy sequences, one works with Cauchy filters (or Cauchy nets). A (respectively, a ) on a uniform space is a filter (respectively, a prefilter) such that for every entourage there exists with In other words, a filter is Cauchy if it contains "arbitrarily small" sets. It follows from the definitions that each filter that converges (with respect to the topology defined by the uniform structure) is a Cauchy filter. A is a Cauchy filter that does not contain any smaller (that is, coarser) Cauchy filter (other than itself). It can be shown that every Cauchy filter contains a unique . The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter. Conversely, a uniform space is called if every Cauchy filter converges. Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology. Complete uniform spaces enjoy the following important property: if is a uniformly continuous function from a dense subset of a uniform space into a complete uniform space then can be extended (uniquely) into a uniformly continuous function on all of A topological space that can be made into a complete uniform space, whose uniformity induces the original topology, is called a completely uniformizable space. A is a complete is a pair consisting of a complete uniform space and a uniform embedding whose image is a dense subset of Hausdorff completion of a uniform space As with metric spaces, every uniform space has a : that is, there exists a complete Hausdorff uniform space and a uniformly continuous map (if is a Hausdorff uniform space then is a topological embedding) with the following property: for any uniformly continuous mapping of into a complete Hausdorff uniform space there is a unique uniformly continuous map such that The Hausdorff completion is unique up to isomorphism. As a set, can be taken to consist of the Cauchy filters on As the neighbourhood filter of each point in is a minimal Cauchy filter,