Pointfree topology version of image of real-valued continuous functions

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.

Pointfree topology version of image of real-valued continuous functions Keywords:

frame , ring of real-valued continuous functions , countable image , f-ring