Non-density of $C^0$-stable mappings on non-compact manifolds

The problem of density of $C^0$-stable mappings is a classical and venerable subject in singularity theory. In 1973, Mather showed that the set of proper $C^0$-stable mappings is dense in the set of all proper mappings, which implies that the set of $C^0$-stable mappings is dense in the set of all mappings if the source manifold is compact. The aim of this paper is to complement Mather's result and to provide new information to the subject. Namely, we show that the set of $C^0$-stable mappings is never dense in the set of all mappings if the source manifold is non-compact. As a corollary of this result and Mather's result, we can obtain a characterization of density of $C^0$-stable mappings, i.e., the set of $C^0$-stable mappings is dense in the set of all mappings if and only if the source manifold is compact. To prove the non-density result, we provide a more essential result by using the notion of topologically critical points.