Irreducible Space is Connected

Theorem
Let $T = \struct {S, \tau}$ be a topological space which is irreducible.

Then $T$ is connected.

Proof
Let $T = \struct {S, \tau}$ be irreducible.

Then:
 * $\forall U_1, U_2 \in \tau: U_1, U_2 \ne \O \implies U_1 \cap U_2 \ne \O$

So trivially there are no two open sets that can form a separation of $T$.

The result follows from definition of connected.

Also see

 * Irreducible Component is Contained in Connected Component
 * Space is Irreducible iff Open Sets are Connected