family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4). Uniform cover definition A uniform space is a set equipped with a distinguished family of coverings called "uniform covers", drawn from the set of coverings of that form a filter when ordered by star refinement. One says that a cover is a star refinement of cover written if for every there is a such that if then Axiomatically, the condition of being a filter reduces to: is a uniform cover (that is, ). If with a uniform cover and a cover of then is also a uniform cover. If and are uniform covers then there is a uniform cover that star-refines both and Given a point and a uniform cover one can consider the union of the members of that contain as a typical neighbourhood of of "size" and this intuitive measure applies uniformly over the space. Given a uniform space in the entourage sense, define a cover to be uniform if there is some entourage such that for each there is an such that These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of as ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other. Topology of uniform spaces Every uniform space becomes a topological space by defining a subset to be open if and only if for every there exists an entourage such that is a subset of In this topology, the neighbourhood filter of a point is This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: and are considered to be of the "same size". The topology defined by a uniform structure is said to be . A uniform structure on a topological space is compatible with the topology if the topology defined by the uniform structure coincides with the original topology. In general several different uniform structures can be compatible with a given topology on Uniformizable spaces A topological space is called if there is a uniform structure compatible with the topology. Every uniformizable space is a completely regular topological space. Moreover, for a uniformizable space the