A topology question on circle $F_{1}$ and $F_{2}$ are two disjoint compact sets in $\mathbb{S}^{1}$, does there exist an arc $\overset{\frown}{AB}$ adjoining $F_{1}$ and $F_{2}$?
Here ''adjoin'' means $A\in F_{1}$ $B\in F_{2}$ and the (open) arc $\overset{\frown}{AB}$ contains no point of $F_{1}\cup F_{2}$.
Could anyone give a proof or a counterexample? I am working on analysis, this is a question related to my research. I would like to know  whether it is true or not.

Yes it's true. Consider the function 
$$\Delta\colon F_1 \times F_2 \to \mathbb{R}$$
defined by restricting the distance function of $S^1$. Since $F_1$ and $F_2$ are compact so is the product, and therefore by the Extreme Value Theorem $\Delta$ has a minimum value $\Delta(x_0, y_0) = d_0$. Since $F_1 \cap F_2 = \emptyset$, $x_0 \neq y_0$. Then you can find your arc from $x_0$ to $y_0$ avoiding $F_1 \cup F_2$, otherwise there would be another pair with a smaller distance.
