Back-and-forth in space: on logics and bisimilarity in closure spaces. Preliminary Extended Version

We adapt the standard notion of bisimilarity for topological models to closure models and refine it for quasi-discrete closure models. We also define an additional, weaker notion of bisimilarity that is based on paths in space and expresses a form of conditional reachability in a way that is reminiscent of Stuttering Equivalence on transition systems. For each bisimilarity we provide a characterisation with respect to a suitable spatial logic. In this report, the detailed proofs of all the results we present are also included.