Pointfree topology version of image of real-valued continuous functions | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 5، دوره 9، شماره 1، مهر 2018، صفحه 59-75 اصل مقاله (644.32 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.29252/cgasa.9.1.59 | ||
| نویسندگان | ||
| Abolghasem Karimi Feizabadi* 1؛ Ali Akbar Estaji2؛ Maryam Robat Sarpoushi3 | ||
| 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. | ||
| چکیده | ||
| 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$. | ||
| کلیدواژهها | ||
| frame؛ ring of real-valued continuous functions؛ countable image؛ $f$-ring | ||
| مراجع | ||
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