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.
Read more about this topic: Totally Bounded Space
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)