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?