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In a classical approach of fibre bundle one always need the cocycle condition is satisfied, namely: $$g_{12} g_{23} g_{31}\equiv 1$$ in $U_1\cap U_2\cap U_3$. However, I do not see why this cocycle condition cannot be succinctly stated as $g_{12} g_{21}=1$. It seems to me that I can still reassemble the original bundle without any problem arises. What is the significance of bringing in the intersection of three sets? I guess maybe I have omitted some important details but I do not see where it is.

My thanks in advance.

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It depends whether you're talking about the cocycle condition for $0$-cochains or for $1$-cochains. Review the construction of transition functions and Čech cochains and coboundaries.

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