Pointfree topology version of image of real-valued continuous functions

Let RL 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 contin- uous functions on a topological space X. Since Cc(X) is the largest subring of C(X) whose elements have countable image, this motivates us to present the pointfree version of Cc(X). The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in RL. In par- ticular, we will introduce the pointfree version of the ring Cc(X). We define a relation from RL into the power set of R, namely overlap. Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as Im(f ) ⊆ S for every continuous function f : X → R and S ⊆ R.