 In the case of vector bundles, this yields formula_12, and indeed this map is an isomorphism (under connect sum of spheres on the right).The above can be generalized by replacing formula_13 and formula_1 with any closed triad formula_15, that is, a space "X", together with two closed subsets "A" and "B" whose union is "X".Then a clutching map on formula_16 gives a vector bundle on "X".