Here we refine the argument of the previous section to prove
is path connected.
There are two main difficulties that arise.
One is that the path connected analogue of Lemma 4.2,
although still true (at least when M is Hausdorff), is much
harder to prove.
The second is that a decreasing intersection of compact path
connected sets need not be path connected, so we can no longer
restrict our attention to the zeros within
.