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The problem of density of $C^0$-stable mappings is a classical and venerablesubject 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 impliesthat the set of $C^0$-stable mappings is dense in the set of all mappings ifthe source manifold is compact. The aim of this paper is to complement Mather'sresult and to provide new information to the subject. Namely, we show that theset of $C^0$-stable mappings is never dense in the set of all mappings if thesource manifold is non-compact. As a corollary of this result and Mather'sresult, 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 ifand only if the source manifold is compact. To prove the non-density result, weprovide a more essential result by using the notion of topologically criticalpoints.

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