Definition:Tangent Bundle

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Defintion

Let $X$ be a differentiable manifold.

The tangent bundle of $X$ is the disjoint union of all the tangent spaces of $X$ at $x$ as the base point $x$ ranges over the entire manifold:

$\displaystyle T \left({X}\right) = \coprod_{x \mathop \in X} T_x \left({X}\right)$