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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:)

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    This seems to be Example 14.5 in Willard's General Topology, [p.93](http://books.google.com/books?id=-o8xJQ7Ag2cC&pg=PA93).2012-07-21
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    What is $Q{}{}$?2012-07-21
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    @Chris Eagle $Q$ denotes all the rational numbers.2012-07-21
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    @Martin Sleziak A little pity.The page 93 of the book cann't be seen on google:)2012-07-21
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    [93](http://i.stack.imgur.com/Dph5H.jpg), [94](http://i.stack.imgur.com/aGEUp.jpg)2012-07-21
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    @Martin Sleziak: Thank you:)2012-07-21

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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.)

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    You meant the proof using BCT from this question: [Application of Baire category theorem in Moore plane](http://math.stackexchange.com/questions/135947/application-of-baire-category-theorem-in-moore-plane)? (Perhaps we should also show that Moore plane is a [Baire space](http://en.wikipedia.org/wiki/Baire_space), so that BCT can be applied. One simple way to see this is to use the fact it contains dense Baire subspace.)2012-07-22
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    @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.2012-07-22
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    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?2012-07-22