Solving a long-standing problem, Brandenburger-Friedenberg-Keisler (2008) gave an epistemic characterization of finite rounds of elimination ofinadmissible---that is, weakly dominated---strategies.\citetalias{BFK08} also formulated rationality and common assumptionof rationality (RCAR), a natural candidate for iterated elimination ofinadmissible strategies in ``complete'' lexicographic type structures.They left as an open question whether RCAR is possible in a completelexicographic type structure.  In this paper, we show that thereexists a large class of complete lexicographic type structures inwhich RCAR is possible and provide a versatile methodology forconstructing them.  Furthermore, we find that the existence of RCAR issensitive to seemingly innocent alterations of the topologies ontypes.  In response, we interpret the topological properties of alexicographic type structure as implicitly describing what it means tohold open-minded beliefs, i.e., beliefs that rational players hold inthe \citetalias{BFK08} framework.