Is the Half Disc Topology normal?

Solution 1:

Unfortunately what you would like to show is false, so I cannot give any hints in that direction. I can give an hint in the opposite direction:

Hint: Consider $x\in L$ and let $P$ be an half-disk neighbourhood of $x$, so that $P$ consists of an open half disk in the upper plane plus the singleton $x$. Note that the closure of $P$ contains all points on the diameter of the open disk. Consider now the complement of $P$, which is a closed set. Can it be separated from $x$ by open neighbourhoods?

Solution 2:

Allessandro has given a correct hint as to why $X$ is not regular (nor $T_3$) but there is a nice related space, the Niemytzki plane or Moore plane (not sure who has priority here) from which we can use the proof of non-normality: $L$ is a closed and discrete subspace of size $\mathfrak{c}=2^{\aleph_0}$, while the whole space $X$ is has a dense subset of size $\aleph_0$. Jones' lemma does the rest: $X$ cannot be normal.

But of course the non-regularity argument is already enough for that. But the Niemytzki plane is a nicer space overall: it's even completely regular and a Moore space.