Recently it was highlighted that one-dimensional antiferromagnetic spin models with frustrated boundary conditions, i.e. periodic boundary conditions in a ring with an odd number of elements, may show very peculiar behavior. Indeed the presence of frustrated boundary conditions can destroy the magnetic order that characterizes such models when different boundary conditions are taken into account and induce novel phase transitions. Motivated by these results, we analyze the effects of the frustrated boundary conditions on several models supporting topological orders. In particular, we focus on the Cluster-Ising model, which presents a symmetry protected topologically ordered phase, and the Kitaev and AKLT chains that, on the contrary, are characterized by a purely topological order. In all these models we find that the different topological orders are not affected by the frustrated boundary conditions. This observation leads naturally to the conjecture that systems supporting topological order are resilient to topological frustration, and thus that topological phases could be identified through this resilience.
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