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This might be probably a simple question, but my wondering was:

Is there any relation between the compactness and the connectedness of a topological space?

Let us consider the specific example (of interest for me) of a subdomain D of a 3D Riemannian manifold.

i) If D is compact, can I say that it is necessarily connected?

ii) If D is simply-connected, can I say that it is compact?

iii) If D is multi-connected, can I say that it is compact?

If one or another answer to these questions is negative, can you please provide me with an example?

Regards

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# Relations between compactness and connectedness

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