Karimi Feizabadi, A., Estaji, A., Robat Sarpoushi, M. (2018). Pointfree topology version of image of real-valued continuous functions. Categories and General Algebraic Structures with Applications, 9(1), 59-75.

Abolghasem Karimi Feizabadi; Ali Akbar Estaji; Maryam Robat Sarpoushi. "Pointfree topology version of image of real-valued continuous functions". Categories and General Algebraic Structures with Applications, 9, 1, 2018, 59-75.

Karimi Feizabadi, A., Estaji, A., Robat Sarpoushi, M. (2018). 'Pointfree topology version of image of real-valued continuous functions', Categories and General Algebraic Structures with Applications, 9(1), pp. 59-75.

Karimi Feizabadi, A., Estaji, A., Robat Sarpoushi, M. Pointfree topology version of image of real-valued continuous functions. Categories and General Algebraic Structures with Applications, 2018; 9(1): 59-75.

Pointfree topology version of image of real-valued continuous functions

^{1}Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.

^{2}Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

^{3}Faculty of Mathematics and Computer Sciences,Hakim Sabzevari University, Sabzevar, Iran.

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 $f:X\rightarrow\mathbb R$ and $ S \subseteq \mathbb R$.

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