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Abstract: Modeled on the book project https://arxiv.org/abs/math/0409584 of Gabber and Ramero, we propose a category of topological sheaves on a not-necessarily noetherian formal scheme which is Quillen-exact, closed symmetric monoidal, and admits cokernels and images. Unless the scheme is affine, the category does not admit kernels. We show that the corresponding (topological) cohomology may be calculated by Cech methods.