Existence of a metric on powerset of a metric space

Question

Let $(X,d)$ be a metric space, let $\mathbb{P}(X)$ denotes the collection of all subsets of $X$ and $\mathbb{K}(X)$ be the collection of all non-empty compact subsets of $X$. It can be shown that $d_\mathbb{H}:\mathbb{K}(X)\times \mathbb{K}(X)\to \mathbb{K}(X)$ defined by $$d_\mathbb{H}(A,B)=(\sup_{x\in A}\inf_{y\in B}d(x,y))\vee(\sup_{x\in B}\inf_{y\in A}d(x,y))$$ is a metric. The metric $d_\mathbb{H}$ is called the Hausdorff metric on $X$ (Should I say Hausdorff metric on $X$ or Hausdorff metric on $\mathbb{K}(X)$?). I was hoping to find some metric $d_\mathbb{P}$ on $\mathbb{P}(X)$ such that on restricting $d_\mathbb{P}$ to $\mathbb{K}(X)$ we get the Hausdorff metric $d_\mathbb{H}$. I am not able to find such a metric. Please help to get such a metric, and if such a metric cannot exist at all, give me the mathematical reasoning behind this.

Answer 1

Pick an arbitrary map $f\colon (\Bbb P(X)\setminus\Bbb K(X))\to \Bbb K(X)$ and define $$ d_{\Bbb P}(A,B):=\begin{cases}d_{\Bbb H}(A,B)&\text{if }A,B\text{ compact}\\ d_{\Bbb H}(f(A),B)+1&\text{if only }B\text{ compact}\\ d_{\Bbb H}(A,f(B))+1&\text{if only }A\text{ compact}\\ d_{\Bbb H}(f(A),f(B))+2&\text{if neither is compact and }A\ne B\\ 0& \text{if }A=B\text{, not compact}\end{cases}$$

You may notice that this is a general way to define a metric on a larger space that coincides with a given metric on a non-empty subspace.