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

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Posts: 12,067
Registered: 7/15/05
Re: Second countable
Posted: Feb 13, 2013 5:49 PM
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Kaba wrote:

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

Just work carefully with the definitions.

There's nothing deep here.

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.

Since the union of the open sets U_x is X, compactness of X
implies there are finitely many points x_1, ..., x_n in X
such that the union of the sets U_(x_1), ..., U_(x_n) is X.

Let B be the union of B_(x_1), ..., B_(x_n). Then B is
a countable collection of open subsets of X. Claim B is
a base for X.

Let V be an open subset of X.

For i = 1, ..., n, let V_i = (V intersect U_(x_i)).

Then V is the union of V_1, ..., V_n and each V_i is a
countable union of open sets from B_(x_i).

It follows that V is a union of open sets from B.

Thus, B is a base for X, hence X is second countable.


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