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Characterisation of strong smooth stability

We begin by recalling Mather’s results on C∞-stability ([5] and [6]). Discussion of stability assumes a topology on C∞(N, P ); Mather’s results use the topology introduced by him in [5] as the Whitney C∞-topology. Following [8], we will denote it τW∞. Theorem 0.1. LetN,P be smooth manifolds (without boundary), f :N→P a proper smooth map. Equip all mapping-spaces with the topology τW∞. Then the following are equivalent.