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quasi
Posts:
12,046
Registered:
7/15/05


Re: Second countable
Posted:
Feb 13, 2013 5:49 PM


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



