totally bounded space

The space X is a totally bounded space if and only if it is a totally bounded set when considered as a subset of itself.

Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded), but the converse is not true in general.

The Hausdorff and Wijsman topologies on Cl(X) coincide if and only if (X, d) is a totally bounded space.

Every subset of a totally bounded space is a totally bounded set; but even if a space is not totally bounded, some of its subsets still will be.

Since the set of the centres of these balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded space is bounded.

(One can also define totally bounded spaces directly, and then define a set to be totally bounded if and only if it is totally bounded when considered as a subspace.)

In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of any fixed "size" (where the meaning of "size" depends on the given context).

But it is no longer true (that is, the proof requires choice) that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice.