Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58539120180701Pointfree topology version of image of real-valued continuous functions597550745ENAbolghasem Karimi FeizabadiDepartment of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.Ali Akbar EstajiFaculty of Mathematics and Computer Sciences,
Hakim Sabzevari University, Sabzevar, Iran.Maryam Robat SarpoushiFaculty of Mathematics and Computer Sciences,Hakim Sabzevari University, Sabzevar, Iran.Journal Article20170318Let $ { 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 $C_c(X).$<br />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 <em>overlap</em>. 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$.http://cgasa.sbu.ac.ir/article_50745_d90d55e08316779860740922b0388294.pdf