This paper introduces a novel combinatorial structure known as the topological s − t cut, a specialized type of s−t cut applicable to acyclic networks. Unlike traditional s−t cuts, topological s−t cuts intersect with exactly one arc of any s−t path d. A linear programming formulation is presented to tackle this problem and it is shown how to solve the problem using the well-known maximum flow algorithms on auxiliary networks. Additionally, a linear-time method is developed for finding minimum topological s−t cuts in s−t planar acyclic networks.