Is $TM \cong M \times \mathbb{R}^n$ as sets? To be clear, I know these sets are not diffeomorphic or even homeomorphic in general. However, I've been told that there doesn't even exist a bijection between these sets.
But suppose $M$ is an $n$-dimensional manifold and let $\{\partial_1|_p, \ldots, \partial_n|_p\}$ be the basis of $T_p M$ with respect to some chart containing $p \in M$. If $v_p \in T_p M$ we have $v_p = v_p^i \partial_i|_p$ for unique real numbers $v_p^i$. Define the function $\lambda: TM \to M \times \mathbb{R}^n$ by $\lambda(p, v_p)=(p, v_p^1, \ldots, v_p^n)$.
Surely this is a well-defined bijection?

Sure: The tangent bundle on $M^n$ is by definition the disjoint union (with a more complicated topology) of the $T_p M$ for $p\in M$, and each $T_p M$ is an $n$-dimensional vector space by definition. Of course, the isomorphisms $T_p M \to \mathbb{R}^n$ are not canonical and can't be made continuous in $p$ for a nontrivial bundle, but that's not a problem if you're just looking for a map of sets.
