[Math] Why compact surfaces can be regarded as region without boundary

differential-geometry

I have been reading DoCarmo and feel quite confused by that he mentioned several times that compact surfaces can be regarded as regions without boundary, which is used in the proof of a corollary of Gauss-Bonnet and several other places.

But I can't figure out why this is the case.

Thanks!

Best Answer

Intuitively, a boundary is like an edge (set of limit points not in the original set). However, compactness says that all limit points are in the original set. So compact surfaces have no boundary. @Zen Lin gave some good examples to hint at this. As embedded objects (speaking intuitively), they are their own boundary within the containing space.

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