I am reading (global) smooth sections of vector bundle. Let me define what do I mean by smooth section.
Let $(E, M, \pi )$ be smooth (real) vector bundle of rank $k$ over smooth manifold $M$ where $\pi : E \rightarrow M$ be smooth surjective (projection) map. A smooth section of $E$ is smooth map $\sigma : M \rightarrow E$ such that $\pi(\sigma(p)) = p$.
I found in wikipedia page Section (Fiber bunlde) that
A section is an abstract characterization of what it means to be a graph. The graph of a smooth function $F : X \rightarrow Y$, where $X$ and $Y$ are smooth real manifolds, can be identified with a function taking its values in the Cartesian product $E = X \times Y$, of $X$ and $Y$: \begin{align} \sigma : X \rightarrow E, \quad \sigma(x) = (x , g(x)) \in E \end{align} Let $\pi : E \rightarrow X $ be the projection onto the first factor: $\pi(x , y) = x$. Then the graph is any function $\sigma$ for which $\pi(\sigma(x)) = x$.
Thus we need to check that $E = X \times Y$ is vector bundle over $X$, because then we can talk about its section. That is first I need to check: for each $x \in X$ the $\pi^{-1}(x) = \{x\} \times Y$ is a real vector space.
At this stage, I am stuck. Someone please explain me how to give real vector space structure on $\{x \} \times Y$.