## A note on the Mackey dual of $C(K)$

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- by Dave Wilkins
- Proc. Amer. Math. Soc.
**118**(1993), 89-92 - DOI: https://doi.org/10.1090/S0002-9939-1993-1128731-5
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## Abstract:

Let $K$ be a compact metric space, and let $\tau$ denote the Mackey topology on $M(K)$ with respect to the $\langle C(K),M(K)\rangle$ duality. That is, $\tau$ is the topology of uniform convergence on the weakly compact subsets of $C(K)$. Just as for the weak$^{\ast }$ topology, the dual space of $(M(K),\tau )$ is $C(K)$. However, $\tau$ is very different from weak$^{\ast }$. Indeed, it is obvious that if $\{ {x_n}\}$ is a sequence converging to $x$ in $K$, then $\delta ({x_n})$ converges to $\delta (x)$ in the weak$^{\ast }$ topology, yet Kirk has shown (Pacific J. Math.**45**(1973), 543-554) that $\{ \delta (x)|x \in K\}$ is closed and discrete in the Mackey topology. We obtain a further result along these lines: For each $A \subset K$ set $\Delta A = \{ \delta (x) - \delta (y)|x \ne y,x,y \in A\}$. Let $\mathcal {D}$ denote the totality of all subsets $A$ of $K$ with the property that $0 \in {\overline {\Delta A} ^\tau }$. Then a closed set is in $\mathcal {D}$ iff it is uncountable. Alternatively stated, a closed subset $A$ of $K$ is countable if and only if there is a weakly compact subset $L$ of $C(K)$ such that for every pair $x,\;y \in A,\;x \ne y$, there is an $h \in L$ with $|h(x) - h(y)| \geqslant 1$.

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## Bibliographic Information

- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**118**(1993), 89-92 - MSC: Primary 46E25; Secondary 46A20, 46J10, 54C35
- DOI: https://doi.org/10.1090/S0002-9939-1993-1128731-5
- MathSciNet review: 1128731