Totally Bounded Space - Examples and Nonexamples

Examples and Nonexamples

• A subset of the real line, or more generally of (finite-dimensional) Euclidean space, is totally bounded if and only if it is bounded. Archimedean property is used.
• The unit ball in a Hilbert space, or more generally in a Banach space, is totally bounded if and only if the space has finite dimension.
• Every compact set is totally bounded, whenever the concept is defined.
• Every totally bounded metric space is bounded. However not every bounded metric space is totally bounded.
• A subset of a complete metric space is totally bounded if and only if it is relatively compact (meaning that its closure is compact).
• In a locally convex space endowed with the weak topology the precompact sets are exactly the bounded sets.
• A metric space is separable if and only if it is homeomorphic to a totally bounded metric space.
• An infinite metric space with the discrete metric (the distance between any two distinct points is 1) is not totally bounded, even though it is bounded.