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Topic: Second countable
Replies: 2   Last Post: Feb 13, 2013 6:13 PM

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Posts: 289
Registered: 5/23/11
Second countable
Posted: Feb 13, 2013 4:55 PM
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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?


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