We consider the p-wave superconductor in one spatial dimension from two perspectives. Firstly as a closed system and secondly as an open system. As a closed system, we analyse the p-wave superconductor as a candidate platform which can host Majorana bound states. In particular, we consider how interaction driven errors could spoil the encoding of protected qubit states constructed from these bound states. We present a generic analysis of how the property of topological quantum order (TQO) suppresses an interaction driven dephasing process. We use this to quantify the extent of correlation between spectral densities from different topological sectors. Our key result, verified numerically for the p-wave superconductor, is that despite mismatched bulk spectra the TQO property ensures that interaction driven phase errors are suppressed up to a time which scales with system length. As an open system, we bring the model into contact with various environments. We do so to understand the interplay of quantum and classical transport phenomena. Our key result in this case is that there is, in general, a strong dependence on the superconductor pairing and the boundary dissipation rates in the final nonequilibrium steady states. Further, we demonstrate that under weak dissipation topological properties of the closed system manifest themselves in a key aspect of the open quantum system dynamics, the Liouvillian gap. We find that this indicator of relaxation time retains information of these properties robustly for a number of dissipative processes.