I started studying Milnor's "Lectures on the H-cobordism theorem". On page 2 he provides the following definitions
Definition 1.3. $(W;V_0,V_1)$ is a smooth manifold triad if $W$ is a compact smooth manifold and $\partial W$ is the disjoint union of two open and closed submanifolds $V_0$ and $V_1$.
My first question is: what exactly is meant by saying "open and closed submanifolds $V_0$ and $V_1$"?
My second question: Everywhere i looked up the definition of a cobordism it was equivalent to Milnor's definition of smooth manifold triads? What am i missing?
A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, $\partial W = M \sqcup N$
Isn't this exactly what milnor defines as a smooth manifold triad?
Milnor then continues with
Definition 1.5 Given two closed smooth $n$-manifolds $M_0$ and $M_1$ (i.e. $M_0,M_1$ compact, $\partial M_0 = \partial M_1 = \emptyset$) a cobordism from $M_0$ to $M_1$ is a $5$-tuple $(W;V_0,V_1;h_0,h_1)$ where $(W;V_0,V_1)$ is a smooth manifold triad and $h_i:V_i \to M_i$ is a diffeomorphism, $i=0,1$.
Am i correct if i conclude "a smooth manifold triad corresponds up to diffeomorphism to a cobordism"? Does this explain the equivalent definitions of smooth manifold triads and cobordisms?
Thanks for any help.