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