Definition 1

A topological space $X$ is called irreducible if every finite intersection of non-empty open sets of $X$ is non-empty.

By considering the empty family of open sets of $X$ it is seen that an irreducible space is non-empty; for a topological space $X$ to be irreducible, it is necessary and sufficient that it be non-empty and that the intersection of two non-empty open sets of $X$ be always non-empty (or, what amounts to the same thing, that the union of two closed sets distinct from $X$ be always distinct from $X$).