Given a manifold with non-empty boundary $\partial M$, I am trying to see if the boundary is connected, will the manifold itself $M$be connected? I can not come up with any counter-example.
2026-03-27 16:09:14.1774627754
If the boundary $\partial M$of a manifold with boundary is connected,is $M$ connected?
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Take the disjoint union of any manifold without boundary and a manifold with a connected boundary like a torus to which you remove a disc.