Noetherian space

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In topology, a Noetherian space is a topological space satisfying the descending chain condition on closed sets.

A closed set in a Noetherian space is again Noetherian with respect to the induced topology.

The motivating example, and origin of the terminology, is that of the Zariski topology on an affine scheme, where the closed sets are precisely the zero sets of ideals of the corresponding ring A (in an order-reversing correspondence). The space is Noetherian if and only if A is a Noetherian ring.