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In the book "Differential forms in Algebraic Topology" by Bott and Tu, the support of a differential form $\omega$ on a manifold $M$ is defined to be "the smallest closed set $Z$ so that $\omega$ restricted to $Z$ is not $0$." (page 24).

I am a little confused, suppose we let $M = \mathbb{R}$ with the trivial atlas $\{ \mathbb{R}, \text{Id} \}$ and consider the $0-$form $\omega = x$. Then any non - zero point would constitute a set on which the restriction of $\omega$ is non - zero. But I expect the authors want the support to be the smallest closed set containing all the points at which $\omega$ is non - zero. Where is my misunderstanding ? Lots of thanks for help!

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up vote 9 down vote accepted

Your expectation is the correct definition: the support of a differential form $\omega \in \Omega^k(M)$ is the set $$\mathrm{supp}(\omega) = \overline{\{p \in M : \omega_p \not\equiv 0\}}.$$

I looked in my copy of Bott and Tu and indeed they defined it incorrectly. Perhaps a better way to phrase their sentence would be "the support of $\omega$ is the smallest closed set $Z$ so that $\omega$ restricted to any point in the interior of $Z$ is not identically $0$," or, as Micah suggests in the comments, "the support of $\omega$ is the smallest closed set $Z$ such that $\omega$ restricted to $M \setminus Z$ is identically $0$."

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I'm gonna guess they meant to write "The smallest closed set $Z$ such that $\omega$ restricted to $Z^c$ is zero." Or at least, that's the closest thing to the original sentence I can think of that's actually true. :-) –  Micah Jul 7 '12 at 23:37
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