  Ketterer and Rajala showed an example of metric measure space, satisfying the
measure contraction property $MCP(0,3)$, that has different topological
dimensions at different regions of the space. In this article I propose a
refinement of that example, which satisfies the $CD(0,\infty)$ condition,
proving the non-constancy of topological dimension for CD spaces. This example
also shows that the weak curvature dimension bound, in the sense of
Lott-Sturm-Villani, is not sufficient to deduce any reasonable non-branching
condition. Moreover, it allows to answer to some open question proposed by
Schultz, about strict curvature dimension bounds and their stability with
respect to the measured Gromov Hausdorff convergence.
