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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 cof 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 armative, 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 ).

On the bounded sets in C c ( X )