Sorry if my question is elementary, I've not touched differential geometry/topology for a while: my question is: is it possible to have a non-compact, embedded submanifold $S$ without boundary of a closed manifold $M?$ If yes, can both $S, M$ be smooth? I'm thinking of a one dimensional such submanifold of the two torus.
Or is there a theorem that says that any non-compact, embedded submanifold $S$ without boundary of a closed manifold $M$ must be closed?