if its Cartesian square is a subset of .) The definition can be extended still further, to any category of spaces with a notion of compactness and Cauchy completion: a space is totally bounded if and only if its (Cauchy) completion is compact. Examples and elementary properties Every compact set is totally bounded, whenever the concept is defined. Every totally bounded set is bounded. A subset of the real line, or more generally of finite-dimensional Euclidean space, is totally bounded if and only if it is bounded. The unit ball in a Hilbert space, or more generally in a Banach space, is totally bounded (in the norm topology) if and only if the space has finite dimension. Equicontinuous bounded functions on a compact set are precompact in the uniform topology; this is the Arzelà–Ascoli theorem. A metric space is separable if and only if it is homeomorphic to a totally bounded metric space. The closure of a totally bounded subset is again totally bounded. Comparison with compact sets In metric spaces, a set is compact if and only if it is complete and totally bounded; without the axiom of choice only the forward direction holds. Precompact sets share a number of properties with compact sets. Like compact sets, a finite union of totally bounded sets is totally bounded. Unlike compact sets, every subset of a totally bounded set is again totally bounded. The continuous image of a compact set is compact. The uniformly continuous image of a precompact set is precompact. In topological groups Although the notion of total boundedness is closely tied to metric spaces, the greater algebraic structure of topological groups allows one to trade away some separation properties. For example, in metric spaces, a set is compact if and only if complete and totally bounded. Under the definition below, the same holds for any topological vector space (not necessarily Hausdorff nor complete). The general logical form of the definition is: a subset of a space is totally bounded if and only if, given any size there exists a finite cover of such that each element of has size at most is then totally bounded if and only if it is totally bounded when considered as a subset of itself. We adopt the convention that, for any neighborhood of the identity, a subset is called () if and only if A subset of a topological group is ()