  Given an infinite topological group G and a cardinal k>0, we say that G is
almost k-free if the set of k-tuples in G^k which freely generate free
subgroups of G is dense in G^k. In this note we examine groups having this
property and construct examples. For instance, we show that if G is a
non-discrete Hausdorff topological group that contains a dense free subgroup of
rank k>0, then G is almost k-free. A consequence of this is that for any
infinite set X, the group of all permutations of X is almost 2^|X|-free. We
also show that an infinite topological group is almost aleph_0-free if and only
if it is almost n-free for each positive integer n. This generalizes the work
of Dixon and Gartside-Knight.
