A minimal regular ring extension of C(X)

M. Henriksen; R. Raphael; R. G. Woods

Fundamenta Mathematicae (2002)

  • Volume: 172, Issue: 1, page 1-17
  • ISSN: 0016-2736

Abstract

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Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring C ( X , τ δ ) of continuous real-valued functions on the space ( X , τ δ ) , where τ δ is the smallest Tikhonov topology on X for which τ ⊆ τ δ and C ( X , τ δ ) is von Neumann regular. The compact and metric spaces for which G ( X ) = C ( X , τ δ ) are characterized. Necessary, and different sufficient, conditions for the equality to hold more generally are found.

How to cite

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M. Henriksen, R. Raphael, and R. G. Woods. "A minimal regular ring extension of C(X)." Fundamenta Mathematicae 172.1 (2002): 1-17. <http://eudml.org/doc/283045>.

@article{M2002,
abstract = {Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring $C(X,τ_δ)$ of continuous real-valued functions on the space $(X,τ_δ)$, where $τ_δ$ is the smallest Tikhonov topology on X for which $τ ⊆ τ_δ$ and $C(X,τ_δ)$ is von Neumann regular. The compact and metric spaces for which $G(X) = C(X,τ_δ)$ are characterized. Necessary, and different sufficient, conditions for the equality to hold more generally are found.},
author = {M. Henriksen, R. Raphael, R. G. Woods},
journal = {Fundamenta Mathematicae},
keywords = {von Neumann regular ring; scattered space; Tikhonov space; RG-space; P-points; Cantor-Bendixson order},
language = {eng},
number = {1},
pages = {1-17},
title = {A minimal regular ring extension of C(X)},
url = {http://eudml.org/doc/283045},
volume = {172},
year = {2002},
}

TY - JOUR
AU - M. Henriksen
AU - R. Raphael
AU - R. G. Woods
TI - A minimal regular ring extension of C(X)
JO - Fundamenta Mathematicae
PY - 2002
VL - 172
IS - 1
SP - 1
EP - 17
AB - Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring $C(X,τ_δ)$ of continuous real-valued functions on the space $(X,τ_δ)$, where $τ_δ$ is the smallest Tikhonov topology on X for which $τ ⊆ τ_δ$ and $C(X,τ_δ)$ is von Neumann regular. The compact and metric spaces for which $G(X) = C(X,τ_δ)$ are characterized. Necessary, and different sufficient, conditions for the equality to hold more generally are found.
LA - eng
KW - von Neumann regular ring; scattered space; Tikhonov space; RG-space; P-points; Cantor-Bendixson order
UR - http://eudml.org/doc/283045
ER -

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