  We develop a sequential-topological study of rational points of schemes of
finite type over local rings typical in higher dimensional number theory and
algebraic geometry. These rings are certain types of multidimensional complete
fields and their rings of integers and include higher local fields. Our results
extend the constructions of Weil over (one-dimensional) local fields. We
establish the existence of an appropriate topology on the set of rational
points of schemes of finite type over any of the rings considered, study the
functoriality of this construction and deduce several properties.
