Counter-example: A non-continuous closed surjection from a normal space onto a non-normal space

Question

Definition: A Hausdorff space is said to be normal if every pair of closed subsets admit disjoint (open) neighborhoods.

Then we have

Theorem: If $X$ is a normal space and $p:X\to Y$ is a closed continuous surjection, then $Y$ is normal.

I want to verify that the hypotheses in the theorem above cannot be weakened. I have already found, relatively easily, counter-examples eliminating (only one at each time!) the "normal", "closed" and "surjective" hypotheses.

But still remains to find a normal space $X$, a non-normal space $Y$, in which exists a surjection $p:X\to Y$ closed but not continuous...

Could you give such an example?

Answer 1

Let $X$ be the Sorgenfrey line (or lower limit topology), which is $\mathbb{R}$ with the topology generated by sets of the form $[a,b)$. Every open set in $\mathbb{R}$ is open in $X$, so the identity map $i:\mathbb{R}^2\to X^2$ is closed. Verify that it is not continuous (easy) and that $X^2$ is not normal (more difficult, but you can find answers on this website).