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p-adic homotopy theory There should be p-adic homotopy theories for every prime p analogous to Sullivan’s Real homotopy theory. A norm on Q induces a unique topology on any finite dimensional vector V space over Q; hence V determines Vp a finite dimensional topological vector space over Qp. If V is infinite dimensional, then use the direct limit of Wp, where W is a finite dimensional subspace of V. If X is a simplicial set and QX={QsX} is the Bousfield-Kan rational homotopy type, then define QpX to be the inverse system of Qp topological vector spaces obtained from QX. This is my proposed definition of p-adic homotopy theory.