Two sets S and T are disjoint and compact in a normed vector space. Define $f(S,T)=\inf\{||s-t||:s \in S, t \in T\}$.

Question

Two sets S and T are disjoint and compact in a normed vector space. Define $f(S,T)=\inf\{||s-t||:s \in S, t \in T\}$. Are there elements $s \in S$ and $t \in T$ s.t. $f(S,T)=||s-t||$?

Answer 1

Consider $X=\mathbb{Q}$ with the usual metric and let $S = \{q \in X: 0 < q < \sqrt{2} \}$ and $T = \{r \in X: \sqrt{2} < r < 2 \}$ then $S$ and $T$ are disjoint compact sets in this metric space but there does not exist $p \in S$ and $r \in T$ such that $||p-r|| = 0$.