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Definition: Let X be a space that is the union of the subspaces X_a, for a in J. The topology of X is said to be coherent with the subspaces X_a provided a subset C of X is closed in X if C intersect X_a is closed in X_a for each a. An equivalent condition is that a set be open in X if its intersection with each X_a is open in X_a.
Source: Topology (second edition) by James R. Munkres