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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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    @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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    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