Constructible set (topology)

A constructable set is a special subset of a topological space and thus an object from topology , a sub-area of ​​mathematics . Constructible sets are mainly considered in algebraic geometry .

definition

Be a topological space. A subset is called constructible if it is a finite union of locally closed subsets. That is, there is a , open subsets and closed subsets with ${\displaystyle X}$ ${\displaystyle A\subseteq X}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle O_{i}\subseteq X}$ ${\displaystyle A_{i}\subseteq X}$

${\displaystyle A=\bigcup _{i=1}^{n}(O_{i}\cap A_{i})}$.

properties

• The constructible sets of a topological space form a Boolean algebra , that is, finite cuts , finite unions and complements of constructible sets can be constructed. This Boolean algebra is precisely the Boolean algebra generated by the open or closed sets.
• Let be topological spaces, a continuous mapping . Then archetypes of constructible subsets can be constructed again under . ${\displaystyle X_{1},X_{2}}$ ${\displaystyle f\colon X_{1}\to X_{2}}$ ${\displaystyle f^{-1}(A)\subseteq X_{1}}$ ${\displaystyle A\subseteq X_{2}}$ ${\displaystyle f}$
• Let be a Noetherian topological space , a constructible subset. Then there is a subset such that it is an open dense subset of the deal . ${\displaystyle X}$ ${\displaystyle Y\subseteq X}$ ${\displaystyle Z\subseteq Y}$ ${\displaystyle Z}$ ${\displaystyle {\bar {Y}}}$
• Constructible sets are compatible with morphisms of algebraic varieties, that is: are algebraic varieties , a morphism of algebraic varieties and a constructible set, then also be constructed. ${\displaystyle X,Y}$ ${\displaystyle \phi \colon X\to Y}$ ${\displaystyle M\subseteq X}$ ${\displaystyle \phi (M)}$

Individual evidence

1. ↑ Borel: Linear Algebraic Groups. 1991, Chapter AG, §1, 1.3.
2. ↑ Borel: Linear Algebraic Groups. 1991, Chapter AG, §1, 1.3.
3. ↑ Borel: Linear Algebraic Groups. 1991, Chapter AG, §1, 1.3.
4. ^ Humphreys: Linear Algebraic Groups. 1975, 4.4 Constructible Sets, Theorem.
5. ^ Harris: Algebraic Geometry. A first course. 1992, Theorem 3.16.

literature

• Joe Harris : Algebraic Geometry. A first course. Springer, New Your 1992, ISBN 3-540-97716-3 , Lecture 3, Constructible sets.
• Armand Borel : Linear Algebraic Groups. 2nd edition, Springer, New York 1991, ISBN 3-540-97370-2 , Chapter AG, §1, 1.3.
• James E. Humphreys : Linear Algebraic Groups. Springer, New York 1975, ISBN 978-1-4684-9445-7 , 4.4 Constructible Sets.