I have a question if it possible, I read some works where they suppose the submanifold be compact and connected, but I don't know the importance of this two properties. For example the compactness can help us to go from local to global, but I don't understand exactly what it means, the same for connected one, I don't know if you have a property that match for compact manifolds and not be true for non compact one. Thank you.
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Do you have any examples where these are used? I know that compactness is really useful since if you cover your space with some open sets then you can pick finitely many that will suffice. This is used, for instance, in the embedding of a cmpct manifold in R^N for some N (see Munkres). Cmpct things are just so much nicer. Connected is the most natural simple case since any manifold (or topological space for that matter) is the disjoint union of its cnctd components and so each connected component is the simplest case, esp. since any continuous function acts independently on each component. – user357980 Jun 13 '17 at 19:24
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See these answers, many of which apply to manifolds. – Dietrich Burde Jun 13 '17 at 19:45
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Thank you fr your answer, I don't know the fact that every open cover admits a finite refinement be so useful? in Rn comapct substes are closed bunded one, but in any manifold and don't see much examples of cmpact submanifolds. – Mohammed Mohammed Jun 15 '17 at 20:53