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.
