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 τ W ∞ . Theorem 0.1. Let N, P be smooth manifolds (without boundary), f : N →P a proper smooth map. Equip all mapping-spaces with the topology τ W ∞ . Then the following are equivalent. (1) f is strongly C ∞ -stable. (2) f is C ∞ -stable. (3) f

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... locally C ∞ -stable. (4) f is infinitesimally stable. Precise definitions of the stability notions mentioned above are given in section 4. Our aim in this paper is to discuss what can be said when f is not proper. Now 0.1 certainly does not hold without some condition on the behaviour of f "at infinity", as Mather was well aware; his counter-examples are to be found in [5] and [6]. We will be concerned with a condition of this kind rather weaker than properness. A smooth map f : usual, (f ) ⊂ N is the set of points at which the tangent map of f is not of rank dim P . In particular, then, (f ) = N if dim N < dim P .) The notion of quasi-properness is introduced in [8]; its interest for us stems from the following result, which is a special case of [8], 4.3.2: Proposition 0.2. Let N, P be smooth manifolds (without boundary), f : N → P a smooth map. If f is strongly C ∞ -stable, then f is quasi-proper.