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).

摘要

如果X是Hausdorff拓扑空间，C _c ( X )是通过赋予X上所有连续函数的代数C ( X )在X的紧子集上一致收敛的拓扑τ _c得到的拓扑代数，则集合Δ ( φ ) := { g ∈ C ( X ) : |g ( x ) | ≤ ϕ ( x ) , x ∈ X } 有界于C _c ( X )，对于每个非负ϕ ∈ C( X )。在本笔记中，我们处理所有此类集合的集合C ⁺是否构成C _c ( X ) 中的有界集合的基的问题。我们给出了答案是积极的例子，以及其他例子，即使集合 Δ( µ )的集合S ^{+ ，具有}µ上半连续，也不能构成这样的基础。尽管如此，我们还是提供了一些情况，包括局部紧致情况，其中S ⁺是C _c ( X ) 中的有界集的基。