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 that is quite different from the notion of a weak topology generated by a set of maps. Let be a topological space and let be a family of subsets of each having the subspace topology. (Typically will be a cover of .) Then is said to be coherent with (or determined by ) if the topology of is recovered as the one coming from the final topology coinduced by the inclusion maps By definition, this is the finest topology on (the underlying set of) for which the inclusion maps are continuous. is coherent with if either of the following two equivalent conditions holds: A subset is open in if and only if is open in for each A subset is closed in if and only if is closed in for each Given a topological space and any family of subspaces there is a unique topology on (the underlying set of) that is coherent with This topology will, in general, be finer than the given topology on A topological space is coherent with every open cover of More generally, is coherent with any family of subsets whose interiors cover As examples of this, a weakly locally compact space is coherent with the family of its compact subspaces. And a locally connected space is coherent with the family of its connected subsets. A topological space is coherent with every locally finite closed cover of A discrete space is coherent with every family of subspaces (including the empty family). A topological space is coherent with a partition of if and only is homeomorphic to the disjoint union of the elements of the partition. Finitely generated spaces are those determined by the family of all finite subspaces. Compactly generated spaces are those determined by the family of all compact subspaces.