where "f" is a continuous real-valued function on "X" and "V" is an entourage of the uniform space R. This uniformity defines a topology, which is clearly coarser than the original topology of "X"; that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any "x" ∈ "X" and a neighbourhood "V" of "x", there is a continuous real-valued function "f" with "f"("x")=0 and equal to 1 in the complement of "V".