Vol. 51, No. 1, 1974

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A characterization of the topology of compact convergence on C(X)

William Alan Feldman

Vol. 51 (1974), No. 1, 109–119

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.

Mathematical Subject Classification 2000
Primary: 46E25
Secondary: 54C40
Received: 16 October 1972
Published: 1 March 1974
William Alan Feldman