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In mathematics , coarse topology is a term in comparison of topologies which specifies the partial order relation of a topological structure to other one -LRB- s -RRB- .. comparison of topologies. comparison of topologies. Specifically , it may refer to  :. Initial topology , the most coarse topology in a certain category of topologies. Initial topology. Initial topology. Trivial topology , the most coarse topology possible on a given set. Trivial topology. Trivial topology