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quasi
Posts:
9,079
Registered:
7/15/05
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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.
quasi
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