In 1948, M. I. Graev proved
that the free topological group on a completely regular Hausdorff space is Hausdorff,
by showing that the free group admits a certain locally invariant Hausdorff
group topology. In 1964, S. Świerczkowski gave a different proof, which also
depends on the construction of a locally invariant topology. Yet another such
construction follows from work of K. Bicknell and S. A. Morris. Graev’s
topology has proved to be essential in the investigation of free products of
topological groups; Świerczkowski’s topology is the key to the work of W. Taylor
on varieties and homotopy laws; and Bicknell and Morris extend results of
Abels on norms on free topological groups. In this paper, the three topologies
are investigated in detail. It is seen that the Graev topology contains the
Świerczkowski topology, which in turn contains that of Bicknell and Morris.
These containments are shown to be proper in general. It is known that the
topology of the free topological group is in general finer than each of these three
topologies.
