Given $(X,d)$ is a metric space. Suppose that $A,B,$ and $C$ are subsets of $X$ which are bounded but non-closed.
One side Hausdorff distance is defined by $d(A,B)= \sup_{x\in A} \inf_{y \in B} d(x,y).$ Does triangle inequality $d(A,B)+d(B,C) \geq d(A,C)$
hold?