On the Dini and Stone–Weierstrass properties in pointfree topology

For a topological space X, it is customary to equip the f  -ring C(X)C(X) or its bounded part C⁎(X)C⁎(X) with the well-known topologies or uniformities of uniform convergence or pointwise convergence, which thank their importance to several fundamental theorems like a.o. the Stone–Weierstrass theorem or Dini's theorem. These theorems are classically proved subject to possible supplementary (often compactness) conditions on X. Alternatively, one can also in many cases characterize exactly those X for which the conclusion of such a theorem holds, i.e. those X that have the Stone–Weierstrass or Dini property. In pointfree topology, for a frame L  , one encounters as a counterpart to C(X)C(X) (resp. C⁎(X)C⁎(X)), the well-studied f  -ring RLRL of real-valued continuous functions on L   and its bounded part R⁎LR⁎L. A pointfree Stone–Weierstrass theorem has been proved in B. Banaschewski [3] and [4]. It is the aim of this note to discuss some topological properties of the f  -ring RLRL or its bounded part which are pointfree counterparts of the Stone–Weierstrass and Dini-type properties for spaces.