I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector is a derivation on the germs of differentiable functions at a point.
Now I am trying to see that the tangent bundle is indeed a differentiable manifold. I get that for an open set $U$ in Eucliden space, it takes the form $U \times \mathbb R^n$ with the vector $\sum a_i x_i$ giving the tangent vector $\sum a_i \partial /\partial x_i$. Now the book says that it is "obvious" that these glue up and give a differentiable manifold.
I may be stupid; but I will be grateful if somebody can hold my hand through this "obvious" statement.
First, prove the following "topological glueing lemma":
Let $M$ be a set, and suppose that $M = \bigcup_\alpha U_\alpha$ for some subsets $U_\alpha$. Suppose moreover that each $U_\alpha$ is a topological space, that $U_\alpha \cap U_\beta$ is open in $U_\alpha$ for each $\alpha, \beta$, and that the two topologies on $U_\alpha \cap U_\beta$ induced by the inclusions $U_\alpha \cap U_\beta \subseteq U_\alpha$ and $U_\alpha \cap U_\beta \subseteq U_\beta$ coincide, for each $\alpha, \beta$. Then there is a unique topology on $M$ in which each $U_\alpha$ is open and such that the restriction of the topology of $M$ to each $U_\alpha$ is the topology of $U_\alpha$.
Then, prove the following "smooth glueing lemma":
Let $M$ be a set, and suppose that $M = \bigcup_\alpha U_\alpha$ for some subsets $U_\alpha$. Suppose moreover that each $U_\alpha$ is a smooth manifold, that $U_\alpha \cap U_\beta$ is open in $U_\alpha$ for each $\alpha, \beta$, and that the two smooth structures on $U_\alpha \cap U_\beta$ induced by the inclusions $U_\alpha \cap U_\beta \subseteq U_\alpha$ and $U_\alpha \cap U_\beta \subseteq U_\beta$ coincide, for each $\alpha, \beta$. Endow $M$ with the topology given by the topological glueing lemma. Then there is a unique smooth structure on $M$ whose restriction to each $U_\alpha$ is the smooth structure of $U_\alpha$.