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P-space∗
CWoo†
2013-03-22 2:46:12
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) = (f 2 + g 2 ).
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 here.
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References
[1] L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand,
(1960).
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