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In this work, we generalize the results obtained in (J. Geom. Anal.,32(6):Paper No.173, 32, 2022), presenting some examples of $CD(0,N)$ spaceshaving different dimensions in different regions, deducing in particular thatthe topological splitting may fail in $CD(0,N)$ spaces. We also observe thatany reasonable non-branching condition may fail in $CD(0,N)$ spaces and thatthe existence of an optimal transport map, between two absolutely continuousmarginals, is not guaranteed by the $CD(0,N)$ condition, without requiring anon-branching assumption. Moreover, we show that the strict $CD(0,N)$ conditionis strictly stronger than the classical $CD(0,N)$ one and it is not stable withrespect to the measured Gromov-Hausdorff convergence.

Examples of $CD(0,N)$ spaces with non-constant dimension