Let $X$ be a finite-dimensional compact metrizable space (these properties might partially be irrelevant; on the other hand, the case $X=[0,1]$ is already interesting to me).
Let $\mathcal F$ be a soft sheaf of $\mathbb Q$-vector spaces on the topology $\mathbb O(X)$ (softness means that restriction of global sections to closed subsets is surjective).
Assume that the dimension of $\mathcal F(X)$ is countable. Question: Is $\mathcal F$ necessarily a direct sum of skyscraper sheaves? If $\mathcal F(X)$ is finite-dimensional then this can be proved using induction on the dimension, I think.
Motivation: I have proved a classification result for certain continuous fields of $C^*$-algebras over $X$. The invariant takes values in soft sheaves (actually in flabby cosheaves, but that is "the same"). Now I am wondering about the range of the invariant.