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 let be a family of subsets of X, each with its induced subspace topology. (Typically C will be a cover of X.) Then X is said to be coherent with C (or determined by C) if the topology of X 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) X for which the inclusion maps are continuous. X is coherent with C if either of the following two equivalent conditions holds: Given a topological space X and any family of subspaces C there is a unique topology on (the underlying set of) X that is coherent with C. This topology will, in general, be finer than the given topology on X.

Examples

Topological union

Let be a family of (not necessarily disjoint) topological spaces such that the induced topologies agree on each intersection Assume further that is closed in X_{\alpha} for each 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 a topological space and 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 ; 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 (inductive) limit (colimit) of in the category Top.

Properties

Let X be coherent with a family of subspaces A function f : X \to Y from X to a topological space Y is continuous if and only if the restrictions are continuous for each This universal property characterizes coherent topologies in the sense that a space X is coherent with C if and only if this property holds for all spaces Y and all functions Let X be determined by a cover Then Let f : X \to Y be a surjective map and suppose Y is determined by For each let be the restriction of f to Then Given a topological space (X,\tau) and a family of subspaces there is a unique topology \tau_C on X that is coherent with C. The topology \tau_C is finer than the original topology \tau, and strictly finer if \tau was not coherent with C. But the topologies \tau and \tau_C induce the same subspace topology on each of the C_\alpha in the family C. And the topology \tau_C is always coherent with C. As an example of this last construction, if C is the collection of all compact subspaces of a topological space (X,\tau), the resulting topology \tau_C defines the k-ification kX of X. The spaces X and kX have the same compact sets, with the same induced subspace topologies on them. And the k-ification kX is compactly generated.