  We study the lattice of T-spaces of a free associative k-algebra over a
nonempty set. It is shown that when the field k is infinite, then the lattice
has a maximum element, and that maximum element is in fact a T-ideal. In
striking contrast, it is then proven that when the field k is finite, the
lattice of T-spaces has infinitely many maximal elements (of which exactly two
are T-ideals). Similar results are also obtained for the free unitary
associative k-algebras. The proof is based on the observation that there is a
natural bijection between the sets of maximal T-spaces of the free associative
$k$-algebras over a nonempty set X and over a singleton set. This permits the
transfer of results from the study of the lattice of T-spaces of the free
associative k-algebra over a one-element set to the general case.
