An irreducible component of an algebraic set is an algebraic subset that is irreducible and maximal (for set inclusion) for this property.
It is a fundamental theorem of classical algebraic geometry that every algebraic set may be written in a unique way as a finite union of irreducible components.
These concepts can be reformulated in purely topological terms, using the Zariski topology, for which the closed sets are the algebraic subsets: A topological space is irreducible if it is not the union of two proper closed subsets, and an irreducible component is a maximal subspace (necessarily closed) that is irreducible for the induced topology.
A topological space X is reducible if it can be written as a union
A topological space is irreducible (or hyperconnected) if it is not reducible.
Equivalently, X is irreducible if all non empty open subsets of X are dense, or if any two nonempty open sets have nonempty intersection.
The empty topological space vacuously satisfies the definition above for irreducible (since it has no proper subsets).
However some authors,[2] especially those interested in applications to algebraic topology, explicitly exclude the empty set from being irreducible.
Every affine or projective algebraic set is defined as the set of the zeros of an ideal in a polynomial ring.
Lasker–Noether theorem implies that every algebraic set is the union of a finite number of uniquely defined algebraic sets, called its irreducible components.
These notions of irreducibility and irreducible components are exactly the above defined ones, when the Zariski topology is considered, since the algebraic sets are exactly the closed sets of this topology.
For this topology, a closed set is irreducible if it is the set of all prime ideals that contain some prime ideal, and the irreducible components correspond to minimal prime ideals.
The number of irreducible components is finite in the case of a Noetherian ring.
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 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.
In this case an irreducible subset is the set of all prime ideals that contain a fixed prime ideal.