Definition:Connected (Topology)/Topological Space/Definition 1

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

$T$ is connected iff it admits no separation.

That is, $T$ is connected iff there exist no open sets $A, B \in \tau$ such that $A, B \ne \varnothing$, $A \cup B = S$ and $A \cap B = \varnothing$.

Also see

 * Equivalence of Definitions of Topological Connectedness