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.

Let X be a topological space and let C = {C_α : 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:

Given a topological space X and any family of subspaces C there is unique topology on (the underlying set of) X which is coherent with C. This topology will, in general, be finer than the given topology on X.

Let ${\displaystyle \{X_{\alpha },\alpha \in A\}}$ be a family of (not necessarily disjoint) topological spaces such that the induced topologies agree on each intersection X_α ∩ X_β. Assume further that X_α ∩ X_β is closed in X_α for each α,β. Then the topological union X is the set-theoretic union

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