# Define a metric on collection of compacts

Let $$(\mathbb{R}^2,d)$$ be a metric space, where $$d$$ is the Euclidean metric on $$\mathbb{R}^2$$. Let $$\kappa$$ denote the set of compact subsets of $$R^2$$. Which one of the following expression defines a metric on $$\kappa$$

1. $$d_1(K_1,K_2) = \inf_{x \in K_1} \inf_{y\in K_2} d(x,y)$$

2. $$d_2(K_1,K_2) = \inf_{x\in K_1} \sup_{y\in K_2} d(x,y)$$

3. $$d_3(K_1,K_2) = \sup_{x\in K_1} \inf_{y\in K_2} d(x,y)$$

4. none of the above.

I know the definition of distance between two sets $$d(A,B) = \inf \{d(x,y) : x \in A,y \in B \}$$ and $$d(A,B)= 0$$ if $$A\cap B \neq \emptyset$$ ; also, $$d(A,B) = 0$$ is possible, even if $$A\cap B = \emptyset$$.

Example Let $$X = \{ \frac{1}{n} : n\in \mathbb{N} \}$$. Let $$A =\{ \frac{1}{2n} : n \in \Bbb N \}$$ and $$B = \{ \frac{1}{2n-1} : n \in \Bbb N \}$$; in this case $$A\cap B =\emptyset$$. I'm not able to define the metric on compact sets. Can someone help me

• Example of what? A and B are not compact. Jun 9 '19 at 11:26
• Yes, A and B are not compact Jun 9 '19 at 12:00

If $$K_1 \subset K_2$$ but $$K_1 \neq K_2$$ then $$d_1(K_1,K_2)=d_3(K_1,K_2)=0$$. Hence $$d_1$$ and $$d_3$$ are not metrics. [One of the requirements for a metric $$d$$ is $$d(x,y)=0$$ implies $$x=y$$]. Similarly, if we take $$K_1=\{x\}$$ and $$K_2=\{x,y\}$$ ($$y \neq x$$) then $$d_2(K_1,K_2)=0$$. Hence none of the three are metrics.