The realistic representation of terrain in hydrological models of fluvial landscapes poses problems particularly for representing convergence and divergence of lateral flows. In this work we consider the topology of contour representation of surfaces. The networks of ridge and drainage lines are linked at critical points: peaks, saddles, confluences, and simple ridge junctions. The properties of these points are given, and from these, three topologic rules are derived. These are used to develop an automatic procedure for generating a catchment boundary and a modified flow net. Use of the flow net increases speed by 2 orders of magnitude over earlier contour-based topographic analyses, and the use of topology increases speed, accuracy, and sophistication over a recent flow net technique. Topology is used to evaluate simplifications of natural surfaces and, in particular, shows the unsuitability of the common ridge-to-stream rectangular flow strip. Finally, implications for the structure of distributed catchment models and for designing and interpreting field experiments are considered.