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

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 Kaba Posts: 289 Registered: 5/23/11
Re: Second countable
Posted: Feb 13, 2013 6:13 PM

14.2.2013 0:49, quasi wrote:
> Kaba wrote:
>

>> Let X be a locally Euclidean Hausdorff space. Show that if X is
>> compact, then X is second countable.
>> ...
>> Any hints?

>
> Since X is locally Euclidean, each x in X is contained in
> an open set U_x homeomorphic to some Euclidean space, hence
> U_x, regarded as a subspace of X, has a countable base,
> B_x say.

--8x--

> It follows that V is a union of open sets from B.
>
> Thus, B is a base for X, hence X is second countable.

My first thought was to take preimages of the basis sets in R^n, to get
a basis for each U_x, and then take the union of these preimage sets
under x. However, this does not work since the homeomorphism distorts
the sets into ever new open sets. Of course, compactness rescues here,