Further Definitions
The definition can be vastly generalized, for example to intersections along subvarieties instead of just at points, or to arbitrary complete varieties.
In algebraic topology, the intersection number appears as the Poincaré dual of the cup product. Specifically, if two manifolds, X and Y, intersect transversely in a manifold M, the homology class of the intersection is the Poincaré dual of the cup product of the Poincaré duals of X and Y.
Read more about this topic: Intersection Number
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