Dynamical symmetry breakings in a two dimensional massless fermion field theory with quartic interactions(the Gross-Neveu model) are investigated in the large fermion number (N)-limit on a space with the S1XS1 topology which may correspond to the finite-volume system at finit-temperature. Four types and the sum of spin-structures are considered. It is shown that the model has a richer phase structure in all boundary conditions than those in R2 or R1 X S1 space-time. It depends on the effective area and the ratio of the circumferences of the two circles whether dynamical symmetry breakings occur or not, where the effective area is of the order of real, area multiplied by the square of the fermion mass, $M_F2$. . In the sum of spin-structures, the phase diagram is the same as that of the periodic-periodic boundary condition and the critical line equation can be written in the modular invariant form. The critical line equations are physical equations which do not depend on renormalization point.