I'm reading the proof of Menger's therom, in Diestel's book, "Graph Theory, 3rd edition". Here's the statement:
Theorem 3.3.1.
Let $G = (V, E)$ be a graph and $A, B \subseteq V $. Then the minimum number of vertices separating $A$ from $B$ in $G$ is equal to the maximum number of disjoint $A–B$ paths in $G$.
As far as I can see, there's a trivial case that is contradictory with this statement. Consider two sets that contains only one node $A=\{a\}$, $B=\{b\}$. Since A itself is a separation of $A, B$, then the the minimum number of vertices separating $A$ from $B$ is no more than 1, but clearly, there might be many disjoint $A-B$ path existing.
I guess it's just a technical problem which can be fixed by slightly changing the definition of "separate" or "$A-B$ path". But what are these fixes?
Another thing about Menger's theorem that I don't get is this corollary in the same book:
Corollary 3.3.5. Let a and b be two distinct vertices of G.
(i) If $ab \notin E$, then the minimum number of vertices $\neq a, b$ separating a from b in G is equal to the maximum number of independent a–b paths in G.
This is said to be a direct result of Theorem 3.3.1, but why do we need the condition that "a" and "b" are NOT adjacent?