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The stability of cosmological event and Cauchy horizons of spacetimes associated with plane symmetric domain walls are studied. It is found that both horizons are not stable against perturbations of null fluids and massless scalar fields; they are turned into curvature singularities. These singularities are light-like and strong in the sense that both the tidal forces and distortions acting on test particles become unbounded when theses singularities are approached.

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