Irreducible topological space

A nonempty topological space is irreducible when it is not the union of two proper closed subsets. Equivalently, every two nonempty open subsets intersect.

A nonempty closed subset which, with its induced topology, is an irreducible topological space. In an affine algebraic set these subsets correspond, with reversed inclusion, to prime ideals of the coordinate ring.

In any topological space, the closure $C$ of a point $x$ is an irreducible closed subset. If $C=A\cup B$ with $A,B$ closed in $C$, then both are closed subsets of the ambient topological space. One contains $x$, so it contains all of $C$ by the definition of closure. Equivalently, $x$ is a generic point of $C$. This does not assert that every irreducible closed subset of an arbitrary topological space has a generic point.

This is the supremum of lengths of strict chains of nonempty irreducible closed subsets. Closedness is relative to the given space. This is the relevant dimension for locally closed subsets of algebraic varieties, rather than a general Hausdorff-space notion of dimension.

The supremum of the lengths of strict chains of nonempty irreducible closed subsets. It is the maximum of the dimensions of the irreducible components. For an affine algebraic set it equals the Krull dimension of its coordinate ring; for a quasi-projective algebraic set it is the supremum of these dimensions over affine open subsets.

An algebraic variety whose irreducible components all have the same dimension of an algebraic set. The components of a generic fibre in the fiber dimension theorem have this property even when that fibre is reducible.

The Krull dimension of ${\mathcal{O}}_{X,x}$ at a point $x$. For a closed point of an irreducible affine variety it equals $\dim X$, by the closed-point dimension lemma for affine domains. At a generic point of a codimension-one subvariety the local dimension is one.

For an irreducible closed subvariety $Y$ of an irreducible variety $X$, its codimension is $\dim X-\dim Y$. In an affine chart it also equals the height of the corresponding prime ideal, by the dimension formula for finite-type domains over a field.

A generic point is a point whose closure is the whole space. An integral scheme has a unique generic point; on an affine scheme associated to an integral domain, it corresponds to the zero prime ideal. Its local ring is the fraction field of the domain. On every nonempty affine chart of an integral scheme these fields identify with the same function field.

An irreducible component is a maximal irreducible closed subset. Every algebraic variety is a finite union of its irreducible components.

The Zariski topology on affine space is Noetherian because polynomial rings are Noetherian. Every closed subset is a finite union of irreducible closed subsets: a minimal counterexample would split into two smaller closed subsets, each already having such a decomposition.