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Suppose $M \subset \mathrm{R}^n$ is a smooth manifold. Does the typical definition of the topology of the tangent bundle $TM$ of $M$ by charts coincide with the topology of $TM$ regarded as a subspace of $\mathrm{R}^{2n}$?

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    Sorry, it really is $\mathrm{R}^{2n}$, as Dylan pointed out2012-06-10

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