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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

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  • $\begingroup$ Example of what? A and B are not compact. $\endgroup$ Commented Jun 9, 2019 at 11:26
  • $\begingroup$ Yes, A and B are not compact $\endgroup$
    – 800123
    Commented Jun 9, 2019 at 12:00

1 Answer 1

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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.

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