guarantees that being "both -close and -close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage there is an entourage that is "not more than half as large". Finally, the last axiom states that the property "closeness" with respect to a uniform structure is symmetric in and A or (or vicinities) of a uniformity is any set of entourages of such that every entourage of contains a set belonging to Thus, by property 2 above, a fundamental systems of entourages is enough to specify the uniformity unambiguously: is the set of subsets of that contain a set of Every uniform space has a fundamental system of entourages consisting of symmetric entourages. Intuition about uniformities is provided by the example of metric spaces: if is a metric space, the sets form a fundamental system of entourages for the standard uniform structure of Then and are -close precisely when the distance between and is at most A uniformity is finer than another uniformity on the same set if in that case is said to be coarser than Pseudometrics definition 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 be a pseudometric on a set The inverse images for can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the is the uniformity defined by the single pseudometric Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces. For a family of pseudometrics on the uniform structure defined by the family is the least upper bound of the uniform structures defined by the individual pseudometrics A fundamental system of entourages of this uniformity is provided by the set of finite intersections of entourages of the uniformities defined by the individual pseudometrics If the family of pseudometrics is finite, it can be seen that the same uniform structure is defined by a single pseudometric, namely the upper envelope of the family. Less trivially, it can be shown that a uniform structure that admits a countable fundamental system of entourages (hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric. A consequence is that any uniform structure can be defined as above by a (possibly uncountable)