Let $X$ be a topological space and $X=X_1 \cup X_2$ with $X_1, X_2$ nonempty open irreducible subsets. Then $X$ is irreducible iff $X_1 \cap X_2 \ne \emptyset$.
The easy part: if it were $X_1 \cap X_2 = \emptyset$ then we would have $$ X = (X \setminus X_1) \cup (X \setminus X_2) $$ and this is impossible since $X$ is irreducible, so it can't be written as a union of two proper closed subsets.
The otherway gives me some problems. Suppose by contradiction $X=C_1 \cup C_2$ with $C_i$ proper closed subsets. Then... what can I do?
Could you please provide any hints, please? Thanks.