complete metric spaces) are often called topologically complete; sometimes the term is also used for Čech-complete spaces or completely uniformizable spaces. Topology See Topological space. Totally bounded A metric space M is totally bounded if, for every r > 0, there exist a finite cover of M by open balls of radius r. A metric space is compact if and only if it is complete and totally bounded. Totally disconnected A space is totally disconnected if it has no connected subset with more than one point. Trivial topology The trivial topology (or indiscrete topology) on a set X consists of precisely the empty set and the entire space X. Tychonoff A Tychonoff space (or completely regular Hausdorff space, completely T3 space, T3.5 space) is a completely regular T0 space. (A completely regular space is Hausdorff if and only if it is T0, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff. U Ultra-connected A space is ultra-connected if no two non-empty closed sets are disjoint. Every ultra-connected space is path-connected. Ultrametric A metric is an ultrametric if it satisfies the following stronger version of the triangle inequality: for all x, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z)). Uniform isomorphism If X and Y are uniform spaces, a uniform isomorphism from X to Y is a bijective function f : X → Y such that f and f−1 are uniformly continuous. The spaces are then said to be uniformly isomorphic and share the same uniform properties. Uniformizable/Uniformisable A space is uniformizable if it is homeomorphic to a uniform space. Uniform space A uniform space is a set X equipped with a nonempty collection Φ of subsets of the Cartesian product X × X satisfying the following axioms: if U is in Φ, then U contains { (x, x) | x in X }. if U is in Φ, then { (y, x) | (x, y) in U } is also in Φ if U is in Φ and V is a subset of X × X which contains U, then V is in Φ if U and V are in Φ, then U ∩ V is in Φ if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in U. The elements of Φ are called entourages, and Φ