"Kuratowski-Mrówka theorem" Need reference

Question

I need a reference "book or paper" for this theorem:

A Hausdorff space $X$ is compact if and only if the projection $P_{Y}:X\times Y\to Y$ is a closed map.

Thank you

The Hausdorff is not needed, actually. Your article also mentions that (you can just use "quasi-compact" and drop the Hausdorff). — Henno Brandsma, Apr 16 '18 at 21:45
The Wikipedia article you quoted, I now see, has many of the same quotations I gave below. — Henno Brandsma, Apr 16 '18 at 21:47

score 1 · Answer 1 · answered Apr 16 '18 at 21:41

If you just want a reasonably recent and trusted reference, I'd quote

Engelking "General Topology", revised and completed edition (1989) Thm. 3.1.16 (the Kuratowski theorem).

In the historical notes Engelking mentions that the 1931 paper

Evaluation de la classe borélienne d'un ensemble de points à l'aide des symboles logiques, Fund. Math 17 (1931), 249-272 by K.Kuratowski

proved the left to right direction for compact metric spaces, after which

Bourbaki Topologie générale ch I et II, Paris (1940)

generalised this to general topological spaces. Finally,

S. Mrówka in Compactness and product spaces, Coll. Math. 7 (1959-1960), 23-25

showed that the closed projection property characterised compact spaces.

If you just want to see a proof, look at my answer here e.g.

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