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).