What is an example of a finite product of closed maps that is not itself a closed map?
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Let $D=\{0\}$ be the one-point discrete space. Let $f:\mathbb{R}\to\mathbb{R}:x\mapsto x$ be the identity map, and let $g:\mathbb{R}\to D:x\mapsto 0$. Both $f$ and $g$ are easily seen to be closed, but $f\times g:\mathbb{R}^2\to\mathbb{R}\times D$ is not: it maps the graph of $xy=1$, which is a closed set in $\mathbb{R}^2$, to $$\Big(\mathbb{R}\setminus\{0\}\Big)\times\{0\}\;,$$ which is not closed in $\mathbb{R}\times D$: $\langle 0,0\rangle$ is in its closure. |
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