Can every manifold be embedded into a compact manifold of the same dimension

Question

Can every connected smooth boundary-less manifold be embedded into a compact smooth boundaryless manifold of the same dimension ? If not, can someone please provide me with a counterexample ? Thank you

Answer 1

There is a counterexample explained in my answer to a related question: an infinite genus surface cannot be embedded into a compact surface.

So for example the Jacob's ladder surface has infinite genus, and hence cannot be embedded into a compact surface.