Pspace
Suppose $X$ is a completely regular^{} topological space^{}. Then $X$ is said to be a Pspace 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 Pspace.
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 vonNeumann 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 Pspace is equivalent to any of the following statements:

•
every zero set^{} is open

•
if $f,g\in C(X)$, then $(f,g)=({f}^{2}+{g}^{2})$.
Some properties of Pspaces:

1.
Every subspace^{} of a Pspace is a Pspace,

2.
Every quotient space^{} of a Pspace is a Pspace,

3.
Every finite product^{} of Pspaces is a Pspace,

4.
Every Pspace has a base of clopen sets.
For more properties of Pspaces, 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  Pspace 

Canonical name  Pspace 
Date of creation  20130322 18:53:13 
Last modified on  20130322 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 