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.