Multidimensional $C^0$ transversality
  In 1994, Sakai introduced the property of $C^0$ 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 $C^0$ transversality of two submanifolds of a topological manifold and a necessary condition of $C^0$ transversality for mappings of metric spaces into $\mathbb{R}^n$. 
Aleksey A. Petrov, Sergei Yu. Pilyugin
math.DS math.GN