P-space
Suppose X is a completely regular topological space
. Then X is said to be a P-space if every prime ideal
in C(X), the ring of continuous functions on X, is maximal.
For example, every space with the discrete topology is a P-space.
Algebraically, a commutative reduced ring R with 1 such that every prime ideal is maximal is equivalent
to any of the following statements:
-
•
R is von-Neumann regular
,
-
•
every ideal in R is the intersection
of prime ideals,
-
•
every ideal in R is the intersection of maximal ideals
,
-
•
every principal ideal
is generated by an idempotent
.
When R=C(X), then R is commutative reduced with 1. In addition to the algebraic characterizations of R above, X being a P-space is equivalent to any of the following statements:
-
•
every zero set
is open
-
•
if f,g∈C(X), then (f,g)=(f2+g2).
Some properties of P-spaces:
-
1.
Every subspace
of a P-space is a P-space,
-
2.
Every quotient space
of a P-space is a P-space,
-
3.
Every finite product
of P-spaces is a P-space,
-
4.
Every P-space has a base of clopen sets.
For more properties of P-spaces, please see the reference below. For proofs of the above properties and equivalent characterizations, see http://planetmath.org/?op=getobj&from=books&id=46here.
References
- 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title | P-space |
---|---|
Canonical name | Pspace |
Date of creation | 2013-03-22 18:53:13 |
Last modified on | 2013-03-22 18:53:13 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 6 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 54E18 |
Classification | msc 16S60 |