A diifficulty in understanding a sentence in a paragraph in Guillemin and Pollack p.77

Question

The paragraph is given below:

But I have a difficulty in understanding the sentence starting in the forth line by "If we furthur ...." until its end, could anyone explain it for me please?

thanks!

Answer 1

This is all basic point-set stuff. $X \cap Z$ is a $0$-dimensional manifold, so it is discrete. If both $X$ and $Z$ are closed then so is $X \cap Z$. If, say, $X$ is compact, then $X \cap Z \subseteq X$. Hence $X \cap Z$ is a closed subset of a compact space, so it is compact. Since it is compact and discrete, it is finite.

Answer 2

If $X$ and $Z$ are closed and at least one of them is compact, then $X\cap Z$ is closed and compact (the intersection of closed sets is closed, and a closed subset of a compact set is compact). Then, the statement is that compact zero-dimensional submanifolds must be finite, which is clear. Zero-dimensional manifolds are discrete, and discrete compact sets are finite.