Vol. 45, No. 2, 1973

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A note on the Mackey topology for (Cb(X), Cb(X))

Ronald Brian Kirk

Vol. 45 (1973), No. 2, 543–554

Let τ denote a completely regular Hausdorff topology on the point set X, let Cb(X) denote the continuous, bounded real-valued functions on X and let Cb(X) denote its Banach dual. If each point of X is identified with the evaluation functional at the point, then X may be treated as a subset of Cb(X). The restriction to X of the Mackey topology for the pair (Cb(X),Cb(X)) will be denoted by μ(τ). The purpose of the paper is to study the topology μ(τ) and its relation to τ. (Obviously, μ(τ) is finer than τ.) It is proved that τ = μ(τ) if and only if τ is discrete. It is shown that μ(τ) is always totally disconnected and that if τ is first countable, then μ(τ) is discrete. An example is given to show that μ(τ) is not discrete in general.

Mathematical Subject Classification 2000
Primary: 46E10
Secondary: 54C40
Received: 15 February 1972
Published: 1 April 1973
Ronald Brian Kirk