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If we have closed subset $F$ of product $X \times Y$ (product topology) does it mean that $p_1(F)$ (projection on first coordinate) is closed in $X$ and $p_2(F)$ in $Y$ are closed? If not, why not (some counterexample or explanation please =)

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This is not always true. A counterexample is $X=Y=\mathbb{R}$, $F=\{(x,y)\in\mathbb{R}^2:xy=1\}$.

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  • $\begingroup$ Thank you =D so simple and yet I couldn't think of counterexample for days... $\endgroup$ – Meow Jun 16 '13 at 21:27
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As you have known, it is not always true. However, putting some conditions on the spaces, we have the following lemma:

If $Y$ is compact, then the projection $\pi_1: X \times Y \to X$ is a closed map.

It means that if $C$ is closed in $X \times Y$, then $\pi_1(C)$ is closed in $X$.

You can find the proof of the lemma here.

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