Date: Feb 13, 2013 4:55 PM
Subject: Second countable
Let X be a locally Euclidean Hausdorff space. Show that if X is compact,
then X is second countable.
The claim can be generalized to if and only if by replacing compact with
sigma-compact, but let's concentrate on this implication. I've managed
to prove X Lindelöf and first-countable, but these seem to be too weak
properties to prove second countability. I'm pretty sure that I should
somehow pull in the second countable basis of R^n by the locally
Euclidean homeomorphisms. Any hints?