  We investigate the notion of relatively amenable topological action and show
that the action of Thompson's group $T$ on $S^1$ is relatively amenable with
respect to Thompson's group $F$. We use this to conclude that $F$ is exact if
and only if $T$ is exact. Moreover, we prove that the groupoid of germs of the
action of $T$ on $S^1$ is Borel amenable.
