# 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 $f,g\in C(X)$, then $(f,g)=(f^{2}+g^{2})$.

Some properties of P-spaces:

1. 1.

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

2. 2.

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

3. 3.

Every finite product 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.

## References

• 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title P-space Pspace 2013-03-22 18:53:13 2013-03-22 18:53:13 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 54E18 msc 16S60