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The Notion of Homotopy,al case. Recall that a path in a space . is simply a map .: . → . The parameter . ∈ . is usually thought of as a measure of time, so that as . runs from 0 to 1 the image point .(.) runs from .(0) to .(1). When .(0) = .(1) = ., say, the path . is called a loop, specifically a loop based at .