The function space of all
continuous real-valued functions on a realcompact topological space X is denoted
by C(X). It is shown that a topology τ on C(X) is a topology of uniform
convergence on a collection of compact subsets of X if and only if (∗)Cr(X)
is a locally m-convex algebra and a topological vector lattice. Thus, the
topology of compact convergence on C(X) is characterized as the finest
topology satisfying (∗). It is also established that if Cτ(X) is an A-convex
algebra (a generalization of locally m-convex) and a topological vector lattice,
then each closed (algebra) ideal in Cτ(X) consists of all functions vanishing
on a fixed subset of X. Some consequences for convergence structures are
investigated.
