Definition (Connected)

$(X,\mathscr{T})$ is connected if, when $A\in \mathbf{2}^{X}$ has the property that $A$ is both open and closed then $$A\in \{ \emptyset,X \}$$

Definition (Disconnected)

If $(X,\mathscr{T})$ is not connected, then it is disconnected, and one can show that it is a disjoint union of connected sets, each of which is called a connected component.