Let $A$ and $B$ be subsets of $X$ and define $d(A, B) = \inf\{d(a, B) | a ∈ A\}.$ Then $ \ d(A, B) = \ d(B, A)$ .True/false

Question

Is the following statement True or False ?

Given (X,d) is a metric space and let $A$ and $B$ be subsets of $X$ and define: $d(A, B) = \inf\{d(a, B) | a ∈ A\}$ . Then $ \ d(A, B) = \ d(B, A)$.

My attempt;

This statement is false. If I take $X= \mathbb{R}$, $A =(0,1)$ and $B= [2,3]$, then a contradiction pops up.

Is my logic correct or not?

Thanks.

Answer 1

The statement is true.

HINT:

Prove that $$\inf\{d(a,B):a \in A\}=\inf\{d(a,b):a \in A,b \in B \}=\inf\{d(b,A):b \in B\}$$

It is not difficult.