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Definition: Let A be a collection of subsets of the space X. A collection B of subsets of X is said to be a refinement of A (or is said to refine A) if for each element D of B, there is an element of C of A containing D. If the elements of B are open sets, we call B an open refinement of A; if they are closed sets, we call B a closed refinement.
Source: Topology (second edition) by James R. Munkres