Again with the late posts…
Now, armed with our two new technical assumptions, we can prove the existence of the refining covers we asserted yesterday.
Since is (now) known to be locally compact, Hausdorff, and second countable, there must exist a countable basis for the topology of with each closure compact. Basically we can start with a neighborhood of each point that has compact closure and whittle it down to a countable basis, using the Hausdorff property to make sure we keep compact closure.
We will construct a sequence of compact sets inductively. Let , which is compact by assumption. Given already defined, let be the first index for which , and define . Then is a sequence of compact sets with , and whose union is all of . Define to be the empty set.
Now, we can write
so for every point we can find a chart sending to and with , for some , and for some .
Indeed, we can surely find some chart around , and intersecting it with some open — which should contain — and with the open — likewise — still gives us a chart. We can subtract off whatever offset we need to make sure that this chart sends to . Then we can take a ball of some radius around and let be its preimage. Scaling up the coordinate map lets us expand this ball until its radius is . Messy, no?
So now the collection of all the preimages as runs over is an open cover of this compact set, and thus it contains a finite subcover, which we write as . Taking the union of all of the gives a countable cover of refining . Each is the domain of a chart with , and the collection of preimages covers , as asserted.
The only thing we haven’t shown here is that is locally finite. But since each point must lie in one of the , so is an open neighborhood of that intersects at most finitely many , and each can intersect at most finitely many , so touches at most finitely many of them itself.
Got all that? We’re not out of the woods yet…