Stability of Closedness of Convex Cones under Linear Mappings

We reconsider the question of when the continuous linear image of a closed convex cone is closed in Euclidean space. In particular, we show that although it is not true that the closedness of the image is preserved under small perturbations of the linear mappings it is "almost" true that the closedness of the image is preserved under small perturbations, in the sense that, for "almost all" linear mappings from

into

\mathbb{R}^m R m

if the image of the cone is closed then there is a small neighbourhood around it whose members also preserve the closedness of the cone.