Irreducible Components
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). The irreducible components are always closed.
Unlike the connected components of a space, the irreducible components need not be disjoint (i.e. they need not form a partition). In general, the irreducible components will overlap. Since every irreducible space is connected, the irreducible components will always lie in the connected components.
The irreducible components of a Hausdorff space are just the singleton sets.
Every non-empty subset of a noetherian topological space can be written as a finite union of irreducible components.
Read more about this topic: Hyperconnected Space
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