A note on the Mackey dual of
Proc. Amer. Math. Soc. 118 (1993), 89-92
Primary 46E25; Secondary 46A20, 46J10, 54C35
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Abstract: Let be a compact metric space, and let denote the Mackey topology on with respect to the duality. That is, is the topology of uniform convergence on the weakly compact subsets of . Just as for the weak topology, the dual space of is . However, is very different from weak. Indeed, it is obvious that if is a sequence converging to in , then converges to in the weak topology, yet Kirk has shown (Pacific J. Math. 45 (1973), 543-554) that is closed and discrete in the Mackey topology. We obtain a further result along these lines: For each set . Let denote the totality of all subsets of with the property that . Then a closed set is in iff it is uncountable. Alternatively stated, a closed subset of is countable if and only if there is a weakly compact subset of such that for every pair , there is an with .
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