Irreducible Space is Locally Connected

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space which is irreducible.

Then $T$ is locally connected.

Proof
Let $T = \left({S, \tau}\right)$ be irreducible.

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

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.