Sheaves

Sheaves are structures on topological spaces that can be described by specifying local data. One should imagine local data as objects that are assigned to open subsets, e.g., the set of all continuous functions defined on the open subset. In order for such an assignment to be considered local data, the objects assigned to two open sets should be compatible with each other when the open sets intersect. For continuous functions, this is the case since restrictions of continuous functions to the intersection are also continuous.