  We study metric measure spaces that have quantitative topological control, as
well as a weak form of differentiable structure. In particular, let $X$ be a
pointwise doubling metric measure space. Let $U$ be a Borel subset on which the
blowups of $X$ are topological planes. We show that $U$ can admit at most $2$
independent Alberti representations. Furthermore, if $U$ admits $2$ Alberti
representations, then the restriction of the measure to $U$ is $2$-rectifiable.
This is a partial answer to the case $n=2$ of a question of the second author
and Schioppa.
