On the bounded sets in C_c (X)

Abstract

If X is Hausdorff topological space and C_c (X) is the topological algebra obtained by endowing the algebra C(X) of all continuous functions on X with the topology τ_c of uniform convergence on the compact subsets of X, then the set Δ(ϕ) := {g ∈ C(X) : |g(x)| ≤ ϕ(x), x ∈ X} is bounded in C_c (X), for every non-negative ϕ ∈ C(X). In this note we deal with the question whether the collection C ⁺ of all such sets constitutes a base of bounded sets in C_c (X). We give instances, where the answer is in the affirmative, and others where even the collection S ⁺ of the sets Δ(µ), with µ upper semi-continuous, fails to constitute such a base. We nevertheless provide situations, including the local compact case, where S ⁺ is a base of bounded sets in C_c (X).