In this work, we analytically derive the caustic equation for a general $N$ -point gravitational lens using methods from algebraic geometry and complex function theory. Based on this equation, we construct the full pre-image of the caustic in the lens plane. This pre-image includes not only the critical curve but also additional closed curves that partition the lens plane into regions mapped to the interior and exterior of the caustic in the source plane. These regions define topological domains within which the number of lensed images remains constant. Notably, when the source moves within a region that does not intersect the caustic, its corresponding images remain confined to specific regions of the lens plane. The effectiveness of the proposed approach is demonstrated using the example of a general binary gravitational lens system.
