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Let $A,B$ be non empty bounded closed subsets of metric space $(X,d)$, define $D(A,B):=\inf\{r:A\subseteq N_r(B) \text{and}B\subseteq N_r(A)\}$. Show that $D$ is a metric on bounded closed subsets of $X$.

I have proved all necessary conditions except the triangle inequality $D(A,B)+D(B,C)\ge D(A,C)$. Please help. Thanks.

JSCB
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1 Answers1

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Note that if $A \subset N_r(B)$ and $B \subset N_s(C)$, then $A \subset N_{r+s}(C)$.

Suppose $A \subset N_r(B)$, $B \subset N_r(A)$ and similarly, $C \subset N_s(B)$, $B \subset N_s(C)$. Then $A \subset N_{r+s}(C)$ and $C \subset N_{r+s}(A)$.

In particular, we have $d(A,C) \le r+s$ for all $r,s$ satisfying the above. Taking the $\inf$ over such $r,s$ yields the desired result.

copper.hat
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