Irreducible Space is Locally Connected

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

Then $T$ is locally 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 $T$.

As a basis consists of open sets, this applies to all sets in a basis for $T$.

The result follows from definition of locally connected.