Let $(A,d)$ be a compact metric space. Then $A$ is connected $ \Longleftrightarrow \forall x,y \in A, \; \forall \varepsilon >0\; \exists \; n \in \mathbb{N}, \; {and } p_1,...,p_n \in A, \; {such that } p_1=x, \; p_n=y \; {and } d(p_i,p_{i+1})<\varepsilon \;(i=1,...,n-1)$