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math8

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Assuming X is normal, second countable. We show there is a homeomorphism of X onto a subspace of [0,1]^w (the Hilbert cube which is metrizable), so X is metrizable.

We first show we can assume that X is infinite.

Suppose X is finite. Since X is normal, it is T1, so every singleton is closed. Hence we can see that every subset of X is closed so every subset is open. So X has the discrete topology. So from here, how can I show that X is metrizable (what would be a homeomorphism between X and [0,1]^w ?).

Now, I can show that X has a countably infinite basis {B1,B2,...} where each Bn is neither X nor the empty set. How do I show that the set of all ordered pairs (Bi,Bk) such that closure(Bi) C Bk is countably infinite?

I can show it's countable because that set is a subset of {B1,B2,...} X {B1,B2,...} which is countable. But how do I show it is infinite?