In mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. It can also be shown to be equivalent that every open subset of such a space is compact, and in fact the seemingly stronger statement that every subset is compact.
Read more about Noetherian Topological Space: Definition, Relation To Compactness, Noetherian Topological Spaces From Algebraic Geometry, Example
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