The proof that Moore plane is not normal I have read was using Cantor's nesting theorem. But I heard that it is also possible to use Baire category theorem to prove and I want to know how.

So, as usually, we start with fomulating two sets

$$Q = \{(x,0):x\in\mathbb{Q}\}$$ and $$P = \{(x,0):x\in\mathbb{P}\} $$

where $\mathbb{Q}$ is rational number and $\mathbb{P}$ is irrational.

Then how to proceed the next step? Any references would be also appreciated.



The proof is by contradiction: We suppose that there are open disjoint sets $U,\, V$ in the Moore plane containing $Q$ and $P$, respectively.

Let $F_n$ be the set of points in $\mathbb P$ for which the ball of radius $1/n$ around $(x,1/n)$ is contained in $V$.

It follows from an application of Baire that there is some $n$ for which $\overline F_n$ has nonempty interior. So there is an open interval $I\subset \overline F_n$. Now pick a rational point in $I$ and show that every open neighborhood of this point intersects $V$ (it intersects some of the balls around $(x,1/n)$).

This contradicts the assumption that $U$ and $V$ are disjoint.


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