Connectedness

Definition (Connectedness)

A topological space is said to be connected if it does not possess a partition into two nonempty open subsets. Otherwise, it is said to be disconnected

Put differently, a topological space $X$ is disconnected iff there exists a pair $U,V$ of disjoint open subsets of $X$ whose union is $X$.

A topological space is connected iff it has no subsets that are both open and closed besides the empty set and the space itself.

Path-Connectedness

If topological space is path-connected, then it is connected.