$[0,1)\times[0,1)$ (lower limit topology) is a regular, but not a normal topological space

Question

Let $X=[0,1)\times[0,1)$, $\tau$ its topology with base $$\beta = \{ [a,b)\times[c,d): 0 \leq a < b \leq 1, 0 \leq c < d \leq 1 \}\;.$$ Please help me prove, that it is regular, but not a normal topological space.

Answer 1

HINTS: For regularity, first prove that $\beta$ is a base for $\tau$ consisting of clopen (i.e., both closed and open) sets; regularity of $X$ follows immediately. For non-normality, let $D=\{\langle x,1-x\rangle:x\in[0,1)\}$; you can easily prove that $D$ is a closed, discrete subset of $X$. Let $H=\{\langle x,1-x\rangle\in D:x\in\Bbb Q\}$, and let $K=D\setminus H$; then $H$ and $K$ are disjoint closed subsets of $X$ that cannot be separated by disjoint open sets. For this part you’ll probably want the Baire category theorem.