A quotient topology of Niemytzki Plane

Question

Let $X$ is the classical Niemytzki plane. We consider the points of the line $y=0$. We paste all the points of $Q$ to one point, and paste all the other points of the line $y=0$ to the other point. Thus we generates a new topological space. It's the quotient topology of Niemytzki Plane. Now my question is this:

Does this new space is still regular? (We know the Niemytzki Plane is a Tychonoff space.)

Thanks ahead:)

This seems to be Example 14.5 in Willard's General Topology, p.93. — Martin Sleziak, Jul 21 '12 at 12:08
@Martin Sleziak A little pity.The page 93 of the book cann't be seen on google:) — Paul, Jul 21 '12 at 12:25

Brian M. Scott · Accepted Answer · 2012-07-22T07:44:07.263

3

It is not even Hausdorff. If $p$ is the point resulting from the identification of$P=\{\langle x,0\rangle:x\in\Bbb R\setminus\Bbb Q\}$, and $q$ is the point resulting from the identification of $Q=\{\langle x,0\rangle:x\in\Bbb Q\}$, then $p$ and $q$ cannot be separated by disjoint open sets in the new space. This follows from the fact that $P$ and $Q$ cannot be separated by disjoint open sets in the Niemytzki plane, as is shown here. (It can also be shown using the Baire category theorem.)

edited Jul 22 '12 at 07:44

answered Jul 21 '12 at 20:19

Brian M. Scott

616,228

You meant the proof using BCT from this question: Application of Baire category theorem in Moore plane? (Perhaps we should also show that Moore plane is a Baire space, so that BCT can be applied. One simple way to see this is to use the fact it contains dense Baire subspace.) – Martin Sleziak Jul 22 '12 at 08:02
@Martin: Yes, that’s essentially the argument that I had in mind. It doesn’t apply the BCT to the Niemytzki plane, though: it uses the fact that $\Bbb R\setminus\Bbb Q$ is a Baire space in the usual topology. – Brian M. Scott Jul 22 '12 at 08:05
Thanks Brain for your answer. Do you mean in your comment that we apply the BCT not to the whole spae, but only to the line $y=0$ with usual topology? – Paul Jul 22 '12 at 11:52

A quotient topology of Niemytzki Plane

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