Real compactness via real maximal ideals of B₁(X)

2021/03/19 by A. Deb Ray, Ray, A. Deb, Atanu Mondal +1
Mathematics · #26A21 #54C30 #54C45 #54C50 #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.2103.10980

openalex publication_date 2021/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, constructing a class of ideals of B₁(X) from proper ideals of C(X) a one-one correspondence between the class of real maximal ideals of C(X) and those of B₁(X) is established. The collection of all real maximal ideals of B₁(X) with hull-kernel topology is proved to be homeomorphic to the space of real maximal ideals of C(X) endowed with a topology finer than the subspace topology induced from its structure space. It is also proved that a Tychonoff space is real compact if and only if every real maximal ideal of B₁(X) is fixed. As a consequence, within the class of real compact T₄ spaces whose points are G_δ, B₁(X) = B₁^*(X) if and only if X is finite.

Related

• Ring-theoretic approaches to point-set topology
• Maximal ideals in rings of real measurable functions
• Influence of ideals in compactifications
• Ideals in Rings and Intermediate Rings of Measurable Functions
• Ideals in CB(X) arising from ideals in X