  We introduce a fractal dimension for a metric space defined in terms of the
persistent homology of extremal subsets of that space. We exhibit hypotheses
under which this dimension is comparable to the upper box dimension; in
particular, the dimensions coincide for subsets of $\mathbb{R}^2$ whose upper
box dimension exceeds $1.5.$ These results are related to extremal questions
about the number of persistent homology intervals of a set of $n$ points in a
metric space.
