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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
.
The lifting lemma below will be used as a substitute for
Lemma 4.2.
Its proof is based on proofs obtained independently by David
desJardins and Emanuel Knill.
Lemma 5.1
(Lifting lemma):
Let M be a Hausdorff space and let 
be
the projection map.
Let 
be a continuous map.
Then there is a continuous map 
such that 
.
SubLemma 5.1
Let 
consists of n copies of a
single point
.
Let 
be an arbitrary function that is a lift
of f.
Then g is automatically continuous at all 
.
[Proof]
SubLemma 5.2
Let 
, 
be closed subintervals of 
such that
is a single point 
.
If 
is a continuous lift of f on 
then
there is a continuous lift g of f on 
such
that 
.
[Proof]
SubLemma 5.3
The conclusions of Sublemma 5.2 hold even if 
and 
intersect in more than a
point.
[Proof]
SubLemma 5.4
If I is a closed subinterval of 
and every 
has a neighborhood on which f has a lift, then f has a lift
on I.
[Proof]
SubLemma 5.5
The same holds if I is any subinterval of 
.
[Proof]
[Proof]
Theorem 5.1
is path connected.
[Proof]
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is path connected