Show that omega_1 is normal?

Asked May 09 '22 at 17:46

Active May 09 '22 at 17:57

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I've already proven the following statement. If $A$ and $B$ are disjoint closed subsets of $\omega_1$, then at least one of them must be bounded. This proof is supposed to be used to prove that $\omega_1$ is normal but I can't figure out how to do that.

I have the following so far: Take A to be the bounded set. Since A is bounded and closed, it is also compact. We have already shown that $\omega_1$ is regular.

edited May 09 '22 at 17:57

Tyrone

17,539

asked May 09 '22 at 17:46

Nicole

What’s definition of $\omega_1$? – user264745 May 09 '22 at 17:52
@user264745 Omega_1 is the first/smallest uncountable ordinal – Nicole May 09 '22 at 17:53
I think the request was for you to explain the topology on $\omega_1$. – Tyrone May 09 '22 at 17:57
I’m Sorry. I don’t known what ordinal is. I just googled it. $\omega_1$ is a well ordered set. You may known, every well ordered set is normal in order topology fact or just check theorem 32.4 of Munkres’ topology. – user264745 May 09 '22 at 18:00
@user264745 it would be the order topology, I believe. – Nicole May 09 '22 at 18:00
If you’re using order topology and you want to show regular($T_1$ and $T_3$), you can prove a stronger result. Link: https://math.stackexchange.com/q/4439856/861687. – user264745 May 09 '22 at 18:08
@user264745 We are showing normal, not regular. – Nicole May 09 '22 at 18:12
@Nicole Yeah. I know that. You can also prove that for normal space. Did my previous of previous comment answer your question? – user264745 May 09 '22 at 18:14

Show that omega_1 is normal?

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