Summary: These notes are part of the first chapter of a series of lectures given by the author in the spring of 1970. The ultimate aim of these notes will be to prove the theorem that the set of topologically stable mappings form a dense subset of \(C^\infty (N, P )\) for any finite dimensional manifolds \(N\) and \(P\), where \(N\) is compact. The first chapter is a study of Thom–Whitney theory of stratified sets and stratified mappings. The connection of the material in these notes with the theorem on the density of topologically stable mappings appears in Section 11, where we give Thom’s second isotopy lemma. This result gives sufficient conditions for two mappings to be topologically equivalent.