Moore plane / Niemytzki plane (Topology)

Question

I am supposed to show that by adding a point to the Moore plane, that the subspace of a $T_4$-space is not necessarily a $T_4$-space. I do know that the Moore plane is NOT a $T_4-$space. Does anybody here know which point could be meant?

Answer 1

Let $X$ be the Niemytzki plane, let $H$ be the open upper half-plane, and let $L$ be the $x$-axis. For each $x\in L$ and $r>0$ let $D(x,r)$ be the closed disk of radius $r$ in the usual topology tangent to $L$ at $x$, and let $\mathscr{D}$ be the set of all such disks. Let $p$ be a point not in $X$, and let $Y=X\cup\{p\}$. Topologize $Y$ be making $X$ an open subset with the Niemytzki topology and taking

$$\left\{Y\setminus\bigcup\mathscr{F}:\mathscr{F}\subseteq\mathscr{D}\text{ is finite}\right\}$$

as a local base at $p$. If $U=X\setminus\bigcup_{k=1}^nD(x_k,r_k)$, where $D(x_1,r_1),\ldots,D(x_n,r_n)\in\mathscr{D}$, is a basic open nbhd of $p$, $s_k>r_k$ for $k=1,\ldots,n$, and $V=X\setminus\bigcup_{k=1}^nD(x_k,s_k)$, then $V$ is also an open nbhd of $p$, and $\operatorname{cl}_YV\subseteq U$. It’s clear that $Y$ is $T_1$, and since $X$ is regular, it follows that $Y$ is $T_3$. $Y$ is also Lindelöf: once $p$ is covered by an open set, only finitely many points of $L$ remain to be covered, and $H$ is Lindelöf. It follows at once that $Y$ is $T_4$.

The idea behind this topology is that counterexamples to normality in $X$ are pairs of infinite subsets of $L$; this topology makes $p$ a limit point of every infinite subset of $L$, thereby killing off all such examples.