On the Dini and Stone-Weierstrass properties in pointfree topology

摘要

For a topological space X, it is customary to equip the f-ring C(X) or its bounded part 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) (resp. C* (X)), the well-studied f-ring RL of real-valued continuous functions on L and its bounded part R*L. A pointfree Stone Weierstrass theorem has been proved in B. Banaschewski [3,4]. It is the aim of this note to discuss some topological properties of the f-ring RL or its bounded part which are pointfree counterparts of the Stone Weierstrass and Dini-type properties for spaces. (C) 2015 Elsevier B.V. All rights reserved.