In Guillemin and Pollack, the stability theorem states that the following classes of smooth maps of a compact manifold X into a manifold Y are stable classes: local diffeomorphisms, immersions, submersions, maps transversal to any specified submanifold ZY , embeddings, and diffeomorphisms. Here a property is stable provided that whenever f0:SY possess the property and ft:XY is a homotopy of f0 , then for some >0, each ft with t< also possesses this property. The collection of maps that possess a particular stable property may be referred to as a stable class of maps.

The proof of this Stability Theorem is given in Guillemin and Pollack. But I want to prove the following: Suppose fs:XY is a family of smooth maps, indexed by a parameter s that varies over a subset S in some Euclidean space. We say that {fs} is a smooth family of mappings if the map F:XSY , defined by F(x,s)=fs(x) , is smooth. Check that the stability theorem generalized immediately to the following: if f0 belongs to any of the classes listed, then there exists >0 such that fs belongs to the same class if s0s< .