Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach that is particularly useful in functional analysis (with pseudometrics provided by seminorms). More precisely, let "f": "X" × "X" → R be a pseudometric on a set "X". The inverse images "U""a" = "f"−1([0,"a"]) for "a" > 0 can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the "U""a" is the uniformity defined by the single pseudometric "f". Certain authors call spaces the topology of which is defined in terms of pseudometrics "gauge spaces".