Irreducible implies connected

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., irreducible space) must also satisfy the second topological space property (i.e., connected space)
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Statement

An irreducible space is a connected space.

Proof

Basically the proof follows directly from the definitions: irreducible means it cannot be written as a union of two proper closed subsets, whereas connected means it cannot be written as a union of two proper disjoint closed subsets.