Rocky Mountain Journal of Mathematics

$P$-spaces and intermediate rings of continuous functions

Will Murray, Joshua Sack, and Saleem Watson

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Abstract

A completely regular topological space $X$ is called a $P$-space if every zero-set in $X$ is open. An intermediate ring is a ring $A(X)$ of real-valued continuous functions on $X$ containing all the bounded continuous functions. In this paper, we find new characterizations of $P$-spaces $X$ in terms of properties of correspondences between ideals in $A(X)$ and $z$-filters on $X$. We also show that some characterizations of $P$-spaces that are described in terms of properties of $C(X)$ actually characterize $C(X)$ among intermediate rings on $X$.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 8 (2017), 2757-2775.

Dates
First available in Project Euclid: 3 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1517648610

Digital Object Identifier
doi:10.1216/RMJ-2017-47-8-2757

Mathematical Reviews number (MathSciNet)
MR3760317

Zentralblatt MATH identifier
06840999

Subjects
Primary: 54C40: Algebraic properties of function spaces [See also 46J10]
Secondary: 46E25: Rings and algebras of continuous, differentiable or analytic functions {For Banach function algebras, see 46J10, 46J15}

Keywords
Rings of continuous functions ideals $P$-spaces $z$-filters regular rings

Citation

Murray, Will; Sack, Joshua; Watson, Saleem. $P$-spaces and intermediate rings of continuous functions. Rocky Mountain J. Math. 47 (2017), no. 8, 2757--2775. doi:10.1216/RMJ-2017-47-8-2757. https://projecteuclid.org/euclid.rmjm/1517648610


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