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During Jacques Lacan’s three meetings with Jean-Pierre Petit, the latter suggested to him that the topological surface of the cross cap did not appear to be the most suitable topological model for representing the R schema, as Lacan had invited psychoanalysts to consider it in his article “On a Question Preliminary to Any Possible Treatment of Psychosis,” which, it is generally agreed, sets out the standard theoretical and clinical paradigm of psychoanalysis. The Boy’s surface that Petit presented to him appeared more appropriate for this topological transformation. This article seeks to illustrate the various modalities and the various stages which preside over the mapping of the R schema onto Boy’s surface, without obscuring the many problems and inconsistencies involved. Logically, an early topologization of the I schema, resulting from a psychotic trigger, concludes the article.