According to one definition at Wiki:
An (n + 1)-dimensional cobordism is a quintuple (W; M, N, i, j) consisting of an (n + 1)-dimensional compact differentiable manifold with boundary, W; closed n-manifolds M, N; and embeddings i : M ⊂ ∂W, j : N ⊂ ∂W with disjoint images.
Note that n-manifolds are supposed to be closed, i.e. compact, without a boundary.
In 1 dimensional case you have a circle and it all makes sense.
However, I often see 0-dimensional manifold (a point) turning via bordism into a 1-dimensional manifold which is isomorphic to a closed interval in R. This 1-dimensional manifold has a border.
Then they construct 2-dimensional bordisms starting from it so what now happened to the definition that M and N are compact and have no border?
Edit based on comments: Is there an expanded cobordism definition where M and N are allowed to have a boundary or a definition that includes corners?