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Using a metric on a metric space $(X,d)$ we can define Hausdorff metric $h$ on the set of all compact subsets of $(X,d)$,say $K(X)$. I am able to show that $h$ is indeed a metric. Suppose instead of metric $d$ on $X$ we had a partial metric $p$ on $X$. If we analogously define Hausdorff partial metric, say $h_p$, similar to the definition of $h$, will $h_p$ be a partial metric non $K(X)$?

Edit

Let $Xne emptyset$. A partial metric on $X$ is a function $p:Xtimes X to mathbb{R}^+$ satisfying

a)$p(x,y)=p(y,x)$ (symmetry)

b)$0le p(x,x) =p(x,y)= p(y,y)$ then $x=y$ (equality)

c)$p(x,x)le p(x,y)$ (small self distances)

d)$p(x,z)le p(x,y)+p(y,z)-p(y,y)$ (triangle inequality)

for all $x,y,zin X.$

Why partial metric is useful?

Some research articles state that, partial metric is useful in modeling partially defined information as comes in computer science.

Definition

$$h_p(A,B)=sup_{ain A}inf_{bin B} p(a,b)$$

$K(X)$: set of all compact subsets in the partial metric space $(X,p)$

Question

Is $h_p$ a partial metric on $K(X)$?