EN
We generalize a major portion of the classical theory of C- and C*-embedded subspaces to pointfree topology, where the corresponding notions are frame C- and C*-quotients. The central results characterize these quotients and generalize Urysohn's Extension Theorem, among others. The proofs require calculations in CL, the archimedean f-ring of frame maps from the topology of the reals into the frame L. We give a number of applications of the central results.