Consider subintervals of $[0,1]$ such that
$E_k$ which has $9^k$ intervals, each of length $10^{-k}$, with total gaps in between the intervals being greater than or equal to $10^{-k}$.
Let $F=\cap_{k=1}E_k$ be non-empty.
Suppose $0<\delta<1$ is such that $\frac{1}{10^{k+1}}\leq \delta <\frac{1}{10^k}$ and that any interval of diameter at most $\delta$ , may intersect 2 intervals. Further suppose each intersected interval contains point of the intersection, $F$, **it follows that the least number of sets in any $\delta$ cover of $F$ is greater than or equal to $\frac{9^k}{2}$.**