Suppose X is a completely regularPlanetmathPlanetmathPlanetmath topological spaceMathworldPlanetmath. Then X is said to be a P-space if every prime idealMathworldPlanetmathPlanetmathPlanetmath 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 commutativePlanetmathPlanetmathPlanetmath reduced ring R with 1 such that every prime ideal is maximal is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to any of the following statements:

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 setPlanetmathPlanetmath is open

  • if f,gC(X), then (f,g)=(f2+g2).

Some properties of P-spaces:

  1. 1.

    Every subspaceMathworldPlanetmathPlanetmath of a P-space is a P-space,

  2. 2.

    Every quotient spaceMathworldPlanetmath of a P-space is a P-space,

  3. 3.

    Every finite productPlanetmathPlanetmath of P-spaces is a P-space,

  4. 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.


  • 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