Let $ω_1$ is the first uncountable ordinal and $ω_1+1$ together with the order topology. Let $U$ and $V$ two disjoint uncountable set consisting of non-limit ordinals, then
- $U$ and $V$ are open and 2. $ω_1$ is the only point in the intersection of their closure.
A limit ordinal is an ordinal number which is neither zero nor a successor ordinal, the other ordinals are isolated or non-limit ordinals. My first question is, what means that $U$ and $V$ consisting of non-limit ordinals?
and why two statement above are true?
If $\alpha$ is some ordinal which is not a limit, then either $\alpha = 0$, which is isolated, or it has a predecessor $\beta$ with $\beta+1 = \alpha$, and in that case $\{\alpha\} = (\beta, \alpha+1)$, which is open in the order topology. So all non-limit ordinals are isolated points, and so sets only containing them are open (as unions of the singletons).
The latter point is not true, as $U$ could contain all even numbers $2n \in \omega$, and the other all $2n+1 \in \omega$, and then $\omega$ would be a common point of their closures.