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Jul 4, 2021 at 21:41 comment added Ryan Budney Generally speaking "irreducible" as a word in topology is pulled directly from algebra. Say you have a class of topological spaces with a monoidal operation, like oriented manifolds (up to diffeo) and the connect-sum operation, or symmetric spaces and the cartesian product operation. Then irreducible just means objects that can't be written as a product, exactly as in algebra.
Jun 4, 2021 at 13:07 history answered C.F.G CC BY-SA 4.0