Irreducible component

A topological space X is reducible if it can be written as a union ${\displaystyle X=X_{1}\cup X_{2}}$ of two 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.

A subset F of a topological space X is called irreducible or reducible, if F considered as a topological space via the subspace topology has the corresponding property in the above sense. That is, ${\displaystyle F}$ is reducible if it can be written as a union ${\displaystyle F=(G_{1}\cap F)\cup (G_{2}\cap F),}$ where ${\displaystyle G_{1},G_{2}}$ are closed subsets of ${\displaystyle X}$, neither of which contains ${\displaystyle F.}$

An irreducible component of a topological space is a maximal irreducible subset. If a subset is irreducible, its closure is also irreducible, so irreducible components are closed.

Every irreducible subset of a space X is contained in a (not necessarily unique) irreducible component of X.[1] Every point of X is contained in some irreducible component of X.

Every affine or projective algebraic set is defined as the set of the zeros of an ideal in a polynomial ring. In this case, the irreducible components are the varieties associated to the minimal primes over the ideal. This is the identification that allows to prove the uniqueness and the finiteness of the decomposition. This decomposition is strongly related with the primary decomposition of the ideal.

In general scheme theory, every scheme is the union of its irreducible components, but the number of components is not necessarily finite. However, in most cases occurring in "practice", namely for all noetherian schemes, there are finitely many irreducible components.

In a Hausdorff space, the irreducible subsets and the irreducible components are the singletons. This is the case, in particular, for the real numbers. In fact, if X is a set of real numbers that is not a singleton, there are three real numbers such that x ∈ X, y ∈ X, and x < a < y. The set X cannot be irreducible since ${\displaystyle X=(X\cap \,]{-\infty },a])\cup (X\cap [a,\infty [).}$

The notion of irreducible component is fundamental in algebraic geometry and rarely considered outside this area of mathematics: consider the algebraic subset of the plane

X = {(x, y) | xy = 0}.

For the Zariski topology, its closed subsets are itself, the empty set, the singletons, and the two lines defined by x = 0 and y = 0. The set X is thus reducible with these two lines as irreducible components.

The spectrum of a commutative ring is the set of the prime ideals of the ring, endowed with the Zariski topology, for which a set of prime ideals is closed if and only if it is the set of all prime ideals that contain a fixed ideal. In this case an irreducible subset is the set of all prime ideals that contain a prime ideal.

This article incorporates material from irreducible on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. This article incorporates material from Irreducible component on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.