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Let $\Gamma$ be the Moore Plane, I have to show that the set $\lbrace (q,0) : q \in \mathbb{Q} \rbrace $ is closed.

I have tried to prove it by trying to find a neigbourhood of every point in the complimentary, but these neigbourhoods aren't contained in the complimentary, since $\mathbb{Q}$ is dense, I don't know how to start to prove it then.

Thanks in advance!

Daniel Brito
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    Hint: For a point $(x,y)$ outside your set, distinguish two cases: (1) $y>0$; (2) $y=0, x\not\in\mathbb Q$. What is the neighbourhood of the points in those two cases which does not intersect your set? In fact, $\mathbb Q$ is a red herring - any subset of the line $y=0$ will be closed by the same argument. –  Dec 01 '20 at 12:21
  • This answer of mine might help... – Henno Brandsma Dec 02 '20 at 22:28

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