Suppose $A,B$ are nonempety subsets of a metrix space $(X,d)$. Define
$$D(A,B) = \inf_{a\in A, \; b \in B } d(a,b) $$
Is this a metrix on $P(X)$? The power set of $X$. My claim is that it is indeed not, but i cannot find a counter example that violates the triangle inequality. Obviously, the other properties a metric must satisfy are obvious in this case. However, i cannot find one that will fail the triangle inequality.
thanks
Best Answer
HINT: While not violating the triangle inequality, here's a reason why this is not a metric. Let $A\cap B\neq\varnothing$, but $A\neq B$. Calculate $D(A,B)$.