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For some reason I am having trouble parsing this bit from Guillemin & Pollack Chapter 2.4:

Let $X, Z$ be transversal closed submanifolds in $Y$ (everything is without boundary). Further, let $X$ be compact. Then, $X \cap Z$ is compact and zero-dimensional. I want to show its finite.

Normally, such proofs should go like if it was infinite then some subsequence satisfying some property would have a limit point, which would be contained in the set and would violate something or the other.

Thanks for the help.

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If I'm not wrong a zero dimensional manifold is a discrete set of points. Then each point is an open set, thus the union of individual points would give you an open cover of the manifold. Compactness then forces the set of points to be finite. –  user27126 Jan 10 '13 at 20:47
    
@Sanchez Ah yes! Thanks! If you would write it out as an answer I'd accept it. –  Apoha Jan 10 '13 at 20:52

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up vote 2 down vote accepted

A zero dimensional manifold is a discrete set of points. Each point is an open set, thus the union of individual points would give you an open cover of the manifold. Compactness then forces the set of points to be finite.

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