Controlled Topological Equivalence of Maps in the Theory of Stratified Spaces and Approximate Fibrations

Ideas from the theory of topological stability of smooth maps are transported into the controlled topological category. For example, the controlled topological equivalence of maps is discussed. These notions are related to the classi cation of manifold approximate brations and manifold strati ed approximate brations. In turn, these maps form a bundle theory which can be used to describe neighborhoods of strata in topologically strati ed spaces.