# Totally Bounded Space - Relationships With Compactness and Completeness

Relationships With Compactness and Completeness

There is a nice relationship between total boundedness and compactness:

Every compact metric space is totally bounded.

A uniform space is compact if and only if it is both totally bounded and Cauchy complete. This can be seen as a generalisation of the Heineâ€“Borel theorem from Euclidean spaces to arbitrary spaces: we must replace boundedness with total boundedness (and also replace closedness with completeness).

There is a complementary relationship between total boundedness and the process of Cauchy completion: A uniform space is totally bounded if and only if its Cauchy completion is totally bounded. (This corresponds to the fact that, in Euclidean spaces, a set is bounded if and only if its closure is bounded.)

Combining these theorems, a uniform space is totally bounded if and only if its completion is compact. This may be taken as an alternative definition of total boundedness. Alternatively, this may be taken as a definition of precompactness, while still using a separate definition of total boundedness. Then it becomes a theorem that a space is totally bounded if and only if it is precompact. (Separating the definitions in this way is useful in the absence of the axiom of choice; see the next section.)