Irreducible not implies Noetherian

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This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property need not satisfy the second topological space property
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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.