  We show that if $\partial\mathcal{R}$ is the boundary of the range of
super-Brownian motion and dim denotes Hausdorff dimension, then with
probability one, for any open set $U$, $\partial\mathcal{R}\cap U\neq\emptyset$
implies $$\text{dim}(U\cap\partial\mathcal{R})=\begin{cases}
4-2\sqrt2\approx1.17&\text{ if }d=2\\ \frac{9-\sqrt{17}}{2}\approx 2.44&\text{
if }d=3. \end{cases}$$ This improves recent results of the last two authors
(arxiv:1711.03486) by working with the actual topological boundary, rather than
the boundary of the zero set of the local time, and establishing a local result
for the dimension.
