Irreducible topological space

The concept of irreducible topological space belongs to the mathematical sub-area of set- theoretical topology , but is mainly used in algebraic geometry .

definition

A non-empty topological space is called irreducible if one and thus all of the following equivalent conditions are met: ${\displaystyle X}$

• ${\displaystyle X}$ is not the union of two closed proper subsets.
• Every two non-empty open subsets of intersect. ${\displaystyle X}$
• Any non-empty open subset of is dense in . ${\displaystyle X}$ ${\displaystyle X}$
• Every open subset of is connected . ${\displaystyle X}$

A subset of a topological space is called irreducible if it is an irreducible space with the induced topology .

properties

• Irreducible spaces are connected.
• Open subsets of irreducible spaces are irreducible.
• A subset of a topological space is irreducible if and only if its closure in is irreducible. ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle {\bar {Y}}}$ ${\displaystyle X}$
• If there is an irreducible space and a continuous mapping, then and is therefore also irreducible. ${\displaystyle X}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle f(X)}$ ${\displaystyle {\overline {f(X)}}}$
• If a point is in any topological space , the closure of the subset in is irreducible, and is a generic point of . ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \{x\}}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle Y}$
• In a Hausdorff space , every irreducible subset consists of a single point.

Irreducible components

The set of irreducible subsets of a topological space is ordered inductively , that is, the union of an ascending chain of irreducible subsets is again irreducible. With the help of Zorn's lemma it then follows that every irreducible set is contained in a maximal irreducible set; such maximal irreducible sets are also called irreducible components . Since closings of irreducible sets are again irreducible, irreducible components must be closed because of their maximality.

Every topological space is the union of its irreducible components, because every point lies in the irreducible set , and this according to the above in an irreducible component. ${\displaystyle x}$ ${\displaystyle \{x\}}$

In a Noetherian space the number of irreducible components is finite. This is important for algebraic geometry , since affine varieties are Noetherian spaces.

Related terms

A topological space is called sober if every irreducible subset has a generic point . If the space also fulfills the axiom of separation T ₀ , then defined ${\displaystyle X}$

${\displaystyle x\mapsto {\overline {\{x\}}}}$

a bijection between points of and closed irreducible subsets of . ${\displaystyle X}$ ${\displaystyle X}$

literature

• Alexander Grothendieck : Éléments de géométrie algébrique . Volume 1: Le langage des schémas (= Publications mathématiques de l'IHÉS. 4). Institut des Hautes Études Scientifiques, Paris 1960, pp. 5–228, ( online (PDF; 27 MB) ).
• Ian G. MacDonald : Algebraic Geometry. Introduction to Schemes (= Mathematics Lecture Note Series. 14, ZDB -ID 790026-0 ). WA Benjamin Inc., New York NY et al. 1968.