Irreducible and Noetherian topological spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 2 June 2011

Noetherian spaces

A non-empty topological space X is irreducible if every pair of non-empty open sets in X intersect (thus X is as far as possible from being Hausdorff). Equivalent conditions:

(a)   X is not the union of two proper closed subsets.
(b)   If Fi (1in) are closed subsets which cover X, then X=Fi for some i.
(c)   Every non-empty open set is dense in X.
(d)   Every open set in X is connected.


(1)   Let X be an infinite set, and topologize X by taking the closed subsets to be X itself and all finite subsets of X. Then X is irreducible.
(2)   Any irreducible algebraic variety with the Zariski topology.

A subset Y of a space X is irreducible if Y is irreducible in the induced topology. The following facts are not hard to prove:

(i)   If (Fi) 1in is a finite closed covering of a space X, and if Y is an irreducible subset of X, then YFi for some i.
(ii)   If X is irreducible, every non-empty open subset of X is irreducible.
(iii)   Let (Ui) 1in be a finite open covering of a space X, the Ui being non-empty. Then X is irreducible if and only if each Ui is irrducible and meets each Uj.
(iv)   If Y is a subset of X, then Y is irreducible if and only if Y is irreducible.
(v)   The image of an irreducible set under a continuous map is irreducible.
(vi)   X has maximal irreducible subsets; they are all closed and they cover X. (Use Zorn's lemma for (vi).)

The maximal irreducible substes of X are called the irreducible components of X. Irreducibility is in some ways analogous to, but stronger than, connectedness.

If xX, then {x} is irreducible and therefore (by (iv) above) so is {x}. If V is an irreducible subset of X and V= {x} for some xX, then x is a generic point of V. If y {x} , y is a specialization of x. The closed set {x} is the locus of x.

A subset Y of a space X is locally closed if Y is the intersection of an open set and a closed set in X, or equivalently if Y is open in its closure Y, or equivalently again if every yY has an open neighborhood Uy in X such that YUy is closed in Uy.

A topological space is Noetherian if the closed subsets of X satisfy the descending chain condition. Equivalent conditions:

(i)   A Noetherian space is quasi-compact.
(ii)   Every subset of a Noetherian space (with the induced topology) is Noetherian.
(iii)   Let X be a topological space and let (Xi) 1in be a finite covering of X. If the Xi are Noetherian, then so is X.
(iv)   If X is Noetherian, the number of irreducible components of X is finite.

The proofs are straightforward.

Notes and References

These notes are taken from [Mac].


[Mac] I.G. Macdonald, Algebraic Geometry: Introduction to Schemes, W.A. Benjamin, New York, 1968.

[Bou] N. Bourbaki, Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp. MR0107661.

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