Irreducible not implies Noetherian

Statement

An irreducible space need not be Noetherian.

Example

Consider a topological space whose underlying set is uncountable, and where the proper closed subsets are precisely the countable subsets (the cocountable topology). The topological space is clearly irreducible, because a union of proper closed subsets is countable, and hence again proper. However, it is not Noetherian, because one can have an infinite descending chain of closed subsets.