The author considers the notions of upper and lower metric dimension, based on covering numbers, introduced by H. Wegmann [J. Reine Angew. Math. 246, 46-75 (1971; Zbl 0208.510)] and D. Kahnert [ibid. 264, 1-28 (1973; Zbl 0266.28009)]. He shows that for each of these, if A is an analytic subset of a Polish space with \(\dim (A)<\infty\), then \(\dim (A)=\dim (K)\) for some compact set \(K\subseteq A\) if and only if dim(A) is equal to the local dimension of A at some point \(p\in A\).