following are equivalent: is a Kolmogorov space is a Hausdorff space is a Tychonoff space for any compatible uniform structure, the intersection of all entourages is the diagonal Some authors (e.g. Engelking) add this last condition directly in the definition of a uniformizable space. The topology of a uniformizable space is always a symmetric topology; that is, the space is an R0-space. Conversely, each completely regular space is uniformizable. A uniformity compatible with the topology of a completely regular space can be defined as the coarsest uniformity that makes all continuous real-valued functions on uniformly continuous. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets where is a continuous real-valued function on and is an entourage of the uniform space This uniformity defines a topology, which is clearly coarser than the original topology of that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any and a neighbourhood of there is a continuous real-valued function with and equal to 1 in the complement of In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space the set of all neighbourhoods of the diagonal in form the unique uniformity compatible with the topology. A Hausdorff uniform space is metrizable if its uniformity can be defined by a countable family of pseudometrics. Indeed, as discussed above, such a uniformity can be defined by a single pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of a vector space is Hausdorff and definable by a countable family of seminorms, it is metrizable. Uniform continuity Similar to continuous functions between topological spaces, which preserve topological properties, are the uniformly continuous functions between uniform spaces, which preserve uniform properties. A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers. Explicitly, a function between uniform spaces is called if for every entourage in there exists an entourage in such that if then or in other words, whenever is an entourage in then is an entourage in , where is defined by All uniformly continuous functions are continuous with respect to the induced topologies. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is