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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. ∗ hPspacei created: h2013-03-2i by: hCWooi version: h41735i Privacy setting: h1i hDefinitioni h54E18i h16S60i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 References [1] L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960). 2