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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.

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  • $\begingroup$ I suppose you want to require the Hausdorff condition? If not, the indiscrete topology seems to work. $\endgroup$ Nov 20 '14 at 14:37
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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.

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  • $\begingroup$ Full citation: Rudin, Mary Ellen. $\aleph$-Dowker spaces. Czechoslovak Math. J. 28 (103) (1978), no. 2, 324–326. MR 478098 $\endgroup$ Nov 20 '14 at 19:33
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    $\begingroup$ although not exactly your question, you (the OP) may want to know about interest in recent years in productively Lindelof spaces, google it or see MSE question math.stackexchange.com/questions/730240/… $\endgroup$
    – Mirko
    Nov 20 '14 at 22:18

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