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?