A note on Ck(irrationals)

The main result in this note was obtained in 2001. It is that Ck(P) does not have a σ-closure-preserving base at the origin consisting of countable unions of the usual basic open sets centered on the origin. A slightly different proof was subsequently published by Gartside and Glyn [1] so this note will not be published unless it is supplemented by new results. Notation. In our context, X is a space homeomorphic to the space of irrational numbers with the usual topology, while Ck(X) denotes the ring of continuous realvalued functions on X, with the compact-open topology. If h ∈ Ck(X) and K is a compact subset of X and ρ>0, let B(h, K, ρ) ={f∈X:|h(x)−f(x)|<ρfor all x ∈ K}. As is well known, these sets form a base for Ck(X). Given a real number r, we let r denote the constant function with domain X and range {r}, and we let − → r denote the constant function with domain Ck(X) and range {r}. Let T denote the space of all nondecreasing functions from ω to itself, with the product topology. As is well known, T is homeomorphic to ω ω itself, and thus to the space P of irrationals. The following two facts are well known, but the proofs are so short that they are included here: Theorem A. Ck(X) is a cosmic space; that is, it has a countable network. Proof. Let B be a countable base for P, and for B ∈Band rationals q<r, let [B,(q, r)] = all f in Ck(P) with f(B) ⊂ (q, r), then the collection of all finite intersections of such things is a network. � Corollary. Ck(X) is hereditarily separable and hereditarily Lindelöf, and sequentially separable. Proof. A space is cosmic iff it is the continuous image of a separable metrizable space. Sequential separability is an immediate consequence, while the other two properties are easy consequences of having a countable network. � This corollary allows us to write any open set of Ck(P) and hence of Cκ(T)as a countable union of basic open sets. Because of the corollary and the following well-known theorem, the question of whether Ck(P) isM1reduces to the question of whether there can be a σ-closure-preserving base of open sets about the origi