Complete uniform spaces enjoy the following important property: if "f": "A" → "Y" is a "uniformly continuous" function from a "dense" subset "A" of a uniform space "X" into a "complete" uniform space "Y", then "f" can be extended (uniquely) into a uniformly continuous function on all of "X". A topological space that can be made into a complete uniform space, whose uniformity induces the original topology, is called a completely uniformizable space. As with metric spaces, every uniform space "X" has a Hausdorff completion: that is, there exists a complete Hausdorff uniform space "Y" and a uniformly continuous map "i": "X" → "Y" with the following property:  The Hausdorff completion "Y" is unique up to isomorphism. As a set, "Y" can be taken to consist of the "minimal" Cauchy filters on "X". As the neighbourhood filter B("x") of each point "x" in "X" is a minimal Cauchy filter, the map "i" can be defined by mapping "x" to B("x"). The map "i" thus defined is in general not injective; in fact, the graph of the equivalence relation "i"("x") = "i"("x" ') is the intersection of all entourages of "X", and thus "i" is injective precisely when "X" is Hausdorff. The uniform structure on "Y" is defined as follows: for each "symmetric" entourage "V" (i.e., such that ("x","y") is in "V" precisely when ("y","x") is in "V"), let "C"("V") be the set of all pairs ("F","G") of minimal Cauchy filters "which have in common at least one V-small set". The sets "C"("V") can be shown to form a fundamental system of entourages; "Y" is equipped with the uniform structure thus defined. The set "i"("X") is then a dense subset of "Y". If "X" is Hausdorff, then "i" is an isomorphism onto "i"("X"), and thus "X" can be identified with a dense subset of its completion. Moreover, "i"("X") is always Hausdorff; it is called the Hausdorff uniform space associated with "X".