The phase behavior of fluid mixtures is understood by the critical lines in fluid-gas diagrams. We investigated the critical lines of polymer-solvent systems at the mathematical double point, where two critical lines meet and exchange branches, and its environment within the framework of a model that combines the lattice gas model of Schouten. ten Seldam and Trappeniers with the Flory-Huggins theory. The critical lines are expressed as a function of x(1) and x(2), the density of type 1 polymer molecules and the density of type 2 polymer molecules, respectively; in this way global phase diagrams are presented and discussed in the density-density plane. Density-density plots are preferable when studying the differences in behavior of different classes, since they enable us to follow the connectivities in a systematic way. In this study the connectivity of critical lines at the mathematical double point and its around is investigated in detail. We also discuss the topology of the critical lines according to the Sadus classification scheme for ternary mixtures. (C) 2010 Elsevier B.V. All rights reserved.