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?