  We prove that for a connected, semisimple linear Lie group $G$ the spaces of
generating pairs of elements or subgroups are well-behaved in a number of ways:
the set of pairs of elements generating a dense subgroup is Zariski-open in the
compact case, Euclidean-open in general, and always dense. Similarly, for
sufficiently generic circle subgroups $H_i$, $i=1,2$ of $G$, the space of
conjugates of $H_i$ that generate a dense subgroup is always Zariski-open and
dense. Similar statements hold for pairs of Lie subalgebras of the Lie algebra
$Lie(G)$.
