Vacuum polarization and spontaneous symmetry breaking in a two-dimensional massless fermion field theory with quartic interactions (the Gross-Neveu model) are studied in a cylindrical (${R}^{1}$\ifmmode\times\else\texttimes\fi{}${S}^{1}$) space-time topology. In the case of untwisted fermion fields the symmetry-breaking behavior is similar to the model in Minkowski space-time except that the location of the effective potential minima depends upon the size of the space. In the case of a twisted fermion field there exists a critical size of the space such that the symmetry breaking occurs only for space larger than the critical one.