Multidimensional C0C0 transversality

In 1994, Sakai introduced the property of C0C0 transversality for two smooth curves in a two-dimensional manifold. This property was related to various shadowing properties of dynamical systems. In this short note, we generalize this property to arbitrary continuous mappings of topological spaces into topological manifolds. We prove a sufficient condition for the C0C0 transversality of two submanifolds of a topological manifold and a necessary condition of C0C0 transversality for mappings of metric spaces into RnRn. Bibliography: 7 titles.