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 of continuous real-valued functions on the space , where is the smallest Tikhonov topology on X for which and is von Neumann regular. The compact and metric spaces for which 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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