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
1Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
2Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
3Faculty 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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