Definition:Tangent Space
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Definition
Let $X$ be a topological space.
Let $\phi: \R^k \to X$ be a mapping such that $\phi \left({0}\right) = x$.
Let the derivative of $\phi$ at $x$ be $d \phi_x$.
Then the tangent space of $X$ at $x$ is defined as the image of $d \phi_0 \left({\R^k}\right)$, and is denoted $T_x \left({X}\right)$.
In this context it needs to be clarified exactly what the derivative of $\phi$ actually means.
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