the question is to prove Urysohn's metrization theorem. But there some steps I need to show first.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Topology question

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**