The Mackey problem for the compact-open topology

Wheeler, Robert F.

doi:10.1090/S0002-9947-1976-0425593-0

The Mackey problem for the compact-open topology
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Trans. Amer. Math. Soc. 222 (1976), 255-265 Request permission

Abstract:

Let ${C_c}(T)$ denote the space of continuous real-valued functions on a completely regular Hausdorff space T, endowed with the compact-open topology. Well-known results of Nachbin, Shirota, and Warner characterize those T for which ${C_c}(T)$ is bornological, barrelled, and infrabarrelled. In this paper the question of when ${C_c}(T)$ is a Mackey space is examined. A convex strong Mackey property (CSMP), intermediate between infrabarrelled and Mackey, is introduced, and for several classes of spaces (including first countable and scattered spaces), a necessary and sufficient condition on T for ${C_c}(T)$ to have CSMP is obtained.

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