Let $(Y,d)$ be a non-degenerate compact metric space, and let $d_H$ be the Hausdorff metric (https://en.wikipedia.org/wiki/Hausdorff_distance) on $K(Y)$ generated by $d$.

Here $K(Y)$ is the set of non-empty compact subsets of $Y$.

Let $M$ be the maximum value of $d_H$.

If $A\in K(Y)$, and $d_H(A,Y)=M$, then is $A$ necessarily nowhere dense in $Y$?.