TITLE: "Productively normal" space
QUESTION [6 upvotes]: If a set $S$ is endowed with the discrete topology $\mathcal{P}(S)$, then for every normal space $N$ the product $S\times N$ is normal.
Question: can we endow a set $S$ with another Hausdorff topology, such that still for all normal spaces $N$ the product $S\times N$ is normal.

REPLY [11 votes]: The answer is no. It was proved by Mary Ellen Rudin in $\aleph$-Dowker spaces (1978) that for any non-discrete Hausdorff space $S$ there is a normal Hausdorff space $N$ such that $S\times N$ is not normal, thus solving in the affirmative Morita´s first conjecture.