Topological sheaves are mathematical structures that combine concepts from topology and algebra. They assign algebraic data, like sets or groups, to open sets of a topological space, ensuring that this assignment is consistent with the topology. This means that if you have a smaller open set within a larger one, the data assigned to the smaller set must relate to the data assigned to the larger set. These sheaves are essential in various fields, including algebraic geometry and sheaf theory. They help in studying local properties of spaces and allow mathematicians to work with global structures by examining local data.