An "'irreducible set "'is a subset of a topological space for which the subspace topology is irreducible.
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Many examples of Noetherian topological spaces come from algebraic geometry, where for the Zariski topology an irreducible set has the intuitive property that any closed proper subset has smaller dimension.
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An "'irreducible component "'in a topological space is a maximal irreducible subset ( i . e . an irreducible set that is not contained in any larger irreducible set ).
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An "'irreducible component "'in a topological space is a maximal irreducible subset ( i . e . an irreducible set that is not contained in any larger irreducible set ).
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Since dimension can only'jump down'a finite number of times, and algebraic sets are made up of finite unions of irreducible sets, descending chains of Zariski closed sets must eventually be constant.
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