Irreducible component

In mathematics, and specifically in algebraic geometry, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation

is the union of the two lines

and

Thus an algebraic set is irreducible if it is not the union of two proper algebraic subsets. It is a fundamental theorem of classical algebraic geometry that every algebraic set is the union of a finite number of irreducible algebraic subsets (varieties) and that this decomposition is unique if one removes those subsets that are contained in another one. The elements of this unique decomposition are called irreducible components.

This notion may be reformulated in topological terms, using Zariski topology, for which the closed sets are the subvarieties: an algebraic set is irreducible if it is not the union of two proper subsets that are closed for Zariski topology. This allows a generalization in topology, and, through it, to general schemes for which the above property of finite decomposition is not necessarily true.

A topological space X is reducible if it can be written as a union ${\displaystyle X=X_{1}\cup X_{2}}$ of two non-empty closed proper subsets ${\displaystyle X_{1}}$, ${\displaystyle X_{2}}$ of ${\displaystyle X}$. A topological space is irreducible (or hyperconnected) if it is not reducible. Equivalently, all non empty open subsets of X are dense or any two nonempty open sets have nonempty intersection.

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