is not the union of two proper closed subsets.
() are closed subsets which cover
, then for
Every non-empty open set is dense in .
Every open set in is connected.
Let be an infinite set, and topologize by
taking the closed subsets to be itself and all finite subsets of
. Then is irreducible.
Any irreducible algebraic variety with the Zariski topology.
A subset of a space is
irreducible if is irreducible in the induced topology.
The following facts are not hard to prove:
is a finite closed covering of a space , and
if is an irreducible subset of ,
then for some .
If is irreducible, every non-empty open subset of is
be a finite open covering of a space , the
being non-empty. Then is irreducible if and only if each
is irrducible and meets each .
If is a subset of , then is irreducible
if and only if is irreducible.
The image of an irreducible set under a continuous map is
has maximal irreducible subsets; they are all closed and they cover . (Use Zorn's lemma for (vi).)
The maximal irreducible substes of are called the irreducible
components of . Irreducibility is in some ways analogous to,
but stronger than, connectedness.
If , then
is irreducible and therefore (by (iv) above) so is
If is an irreducible subset of and
for some , then
is a generic point of . If
, is a specialization of .
The closed set
is the locus of .
A subset of a space is
locally closed if is the intersection of an
open set and a closed set in , or equivalently if
is open in its closure , or equivalently
again if every has an open neighborhood
in such that
is closed in
A topological space is Noetherian if the closed subsets of
satisfy the descending chain condition. Equivalent conditions:
- The open sets in satisfy the ascending chain condition;
- Every open subset of is quasi-compact (i.e. compact but not
- Every subset of is quasi-compact.
A Noetherian space is quasi-compact.
Every subset of a Noetherian space (with the induced topology) is Noetherian.
Let be a topological space and let
be a finite covering of . If the
are Noetherian, then so is .
If is Noetherian, the number of irreducible components of
The proofs are straightforward.
Notes and References
These notes are taken from [Mac].
Algebraic Geometry: Introduction to Schemes,
W.A. Benjamin, New York, 1968.
Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques,
Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp.