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.
Definition
Let X be a topological space and letC = be a family of topological spaces, as subsets of X. Then X is said to be coherent with C if X has the final topology coinduced by the inclusion maps By definition, this is the finest topology on X for which the inclusion maps are continuous. If C is a cover of X, then X is coherent with C if either of the following two equivalent conditions holds:
A subset U is open in Xif and only ifU ∩ Cα is open in Cα for each α ∈ A.
A subset U is closed inX if and only if U ∩ Cα is closed in Cα for each α ∈ A.
The above is not true if C does not cover X Given a topological space X and any family of subspaces C there is unique topology on X that is coherent with C. This topology will, in general, be finer than the given topology on X.
Examples
A topological space X is coherent with every open cover of X.
A topological space X is coherent with every locally finite closed cover of X.
Compactly generated spaces are those determined by the family of all compact subspaces.
A CW complexX is coherent with its family of n-skeletons Xn.
Topological union
Let be a family of topological spaces such that the induced topologies agree on each intersectionXα ∩ Xβ. Assume further that Xα ∩ Xβ is closed in Xα for each α,β. Then the topological union X is the set-theoretic union endowed with the final topology coinduced by the inclusion maps. The inclusion maps will then be topological embeddings and X will be coherent with the subspaces. Conversely, if X is coherent with a family of subspaces that cover X, then X is homeomorphic to the topological union of the family. One can form the topological union of an arbitrary family of topological spaces as above, but if the topologies do not agree on the intersections then the inclusions will not necessarily be embeddings. One can also describe the topological union by means of the disjoint union. Specifically, if X is a topological union of the family, then X is homeomorphic to the quotient of the disjoint union of the family by the equivalence relation for all α, β in A. That is, If the spaces are all disjoint then the topological union is just the disjoint union. Assume now that the set A is directed, in a way compatible with inclusion: whenever . Then there is a unique map from to X, which is in fact a homeomorphism. Here is the direct limit of in the category Top.
Properties
Let X be coherent with a family of subspaces. A map f : X → Y is continuous if and only if the restrictions are continuous for each α ∈ A. This universal property characterizes coherent topologies in the sense that a spaceX is coherent with C if and only if this property holds for all spaces Y and all functions f : X → Y. Let X be determined by a cover C =. Then
If D is a refinement of C and each Cα is determined by the family of all Dβ contained in Cα then X is determined by D.
Let X be determined by and let Y be an open or closed subspace of X. Then Y is determined by. Let X be determined by and let f : X → Y be a quotient map. Then Y is determined by. Let f : X → Y be a surjective map and suppose Y is determined by. For each α ∈ A let be the restriction of f to f−1. Then
If f is continuous and each fα is a quotient map, then f is a quotient map.
f is a closed map if and only if each fα is closed.
