Coherent topology
In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion which is quite different from the notion of a weak topology generated by a set of maps.
Definition
Let X be a topological space and let C = {Cα : α ∈ A} be a family of subspaces of X (typically C will be a cover of X). Then X is said to be coherent with C (or determined by C) if X has the final topology coinduced by the inclusion maps
By definition, this is the finest topology on (the underlying set of) X for which the inclusion maps are continuous.
Equivalently, X is coherent with C if either of the following two equivalent conditions holds:
A subset U is open in X if and only if U ∩ Cα is open in Cα for each α ∈ A.
A subset U is closed in X if and only if U ∩ Cα is closed in Cα for each α ∈ A.