Hausdorff Space that is a non Normal Hausdorff Space

Question

Can someone give me an example of a Hausdorff space (i.e $T_2$), that is not a normal Hausdorff space (i.e $T_4$)?

Answer 1

There are many examples. One simple one is the $K$-topology on $\Bbb R$. Let $K=\left\{\frac1n:n\in\Bbb Z^+\right\}$. Let

$$\mathscr{B}=\{(a,b):a,b\in\Bbb R\text{ and }a<b\}\cup\{(a,b)\setminus K:a,b\in\Bbb R\text{ and }a<b\}\;;$$

then $\mathscr{B}$ is a base for the $K$-topology. This topology is finer than the usual one, so it is certainly Hausdorff. It is not regular, because in this topology $K$ is a closed set, $0\notin K$, and $0$ and $K$ cannot be separated by disjoint open sets.

Another fairly simple example is the Sorgenfrey plane. This post from Dan Ma’s Topology Blog gives much information on it, including a proof that it is not normal.