Good question, but no. For example the Klein bottle or real projective plane are closed surfaces containing no boundary. Incidentally the Mobius strip need not

Good question, but no. For example the Klein bottle or real projective plane are closed surfaces containing no boundary. Incidentally the Mobius strip need not have a edge; you can leave out the boundary circle at the cost of having a non-compact space. This is exactly analogous to the difference between the closed and open intervals: [0, 1] vs (0, 1). In fact the projective plane can be obtained by taking a disc and glueing its boundary to that of the Mobius strip, which makes the two boundaries "cancel".