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 Heine–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.)
Read more about this topic: Totally Bounded Space
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