Definition 3.1:

A subset $Y$ of a topological space $X$ is said to be disconnected if there are non-empty subsets $A$ and $B$ of $X$ such that

$$Y = A\cup B,\quad \overline{A}\cap B = \emptyset,\quad A\cap\overline{B} = \emptyset.$$

If $Y$ is not disconnected we say that $Y$ is connected.