The product of limit point compact Hausdorff spaces is not limit point compact

Question

Let $X, Y$ be limit point compact Hausdorff spaces (to be clear, a space is said to be limit point compact if every infinite subset of it has a limit point). Is it true that $X \times Y$ is limit point compact? The answer should be negative but I can't find a counterexample. Can you?

Answer 1

Limit point compactness is equivalent to countable compactness in $T_1$ spaces, so it suffices to find a pair of countably compact $T_1$ spaces whose product is not countably compact. In this answer you’ll find a reference to the first published example of such a pair, together with a brief description; the spaces in question are not just $T_1$, but Tikhonov, being subspaces of the compact Hausdorff space $\beta\Bbb N$.