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I have found many sources saying that the Lebesgue covering dimension of a (topological or smooth) manifold is the same as the dimension of the manifold. Does anyone know where I can find the proof?

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In case anyone is interested: The book "Infinite-dimensional topology" by J. van Mill proves that the Lebesgue covering dimension agrees with the (small) inductive dimension for a large class of spaces, including both topological and smooth manifolds. An easy induction argument shows that the small inductive dimension agrees with the usual dimension of a manifold.

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