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 X for which the inclusion maps are continuous.
Equivalently, X is coherent with C if either of the following 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.
Given a topological space X and any family of subspaces C there is unique topology on X which is coherent with C. This topology will, in general, be finer than the given topology on X.
Read more about this topic: Coherent Topology
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