Roughly speaking, the N-Archimedean spaces for N∈N are metric spaces which are the opposite of the classical non-Archimedean ones. We prove that a compact N-Archimedean space has no more than N−1 isolated points, is infinite and (N+1)-Archimedean. Moreover, there are compact (N+1)-Archimedean spaces which are not N-Archimedean for every N∈N. Afterwards, we study the dynamics of the expansive systems on N-Archimedean spaces.