Pointfree topology version of image of real-valued continuous functions

Abolghasem Karimi Feizabadi,Ali Akbar Estaji,Maryam Robat Sarpoushi

doi:10.29252/cgasa.9.1.59

Abstract

Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree version of $C_c(X).$The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in $ {mathcal{R}} L$. In particular, we will introduce the pointfree version of the ring $C_c(X)$. We define a relation from $ {mathcal{R}} L$ into the power set of $mathbb R$, namely overlap . Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as $mathop{hbox{Im}} (f) subseteq S$ for every continuous function $f:Xrightarrowmathbb R$ and $ S subseteq mathbb R$.

Highlights

As is well known, C(X) denotes the ring of all real-valued continuous functions on a topological space X
The book Rings of Continuous Functions written by Gillman and Jerison is the best reference to study the
We introduce the pointfree version of image of real-valued continuous functions in the ring of real-valued continuous functions on a frame, namely, RλL

Summary

Introduction

C(X) denotes the ring of all real-valued continuous functions on a topological space X. The ring of real-valued continuous functions on a frame, that is, RL, as the pointfree version of the ring C(X), has been studied prior to 1996 by some authors such as R.N. Ball and A.W. Hager in [1]. A systematic and indepth study of the ring of real continuous functions in pointfree topology was undertaken by B. We introduce the pointfree version of image of real-valued continuous functions in the ring of real-valued continuous functions on a frame, namely, RλL.

Preliminaries

Overlap and its properties

The ring RλL