In a lecture series, I have come across the statement that the topology on the tangent space $TM$ is given by the coarsest topology which makes the projection map $\pi: TM \mapsto M$ continuous.
In the light of the following property (stated below) this statement seems false to me and I would be glad if someone could point out my misunderstanding: The property I am refering to is the following (taken e.g. from Lee: Smooth Manifolds): For each point $p \in M$ there is a neighbourhood $U$ of $p$ such that there is a homeomorphism $\Phi$: $$\Phi: \pi^{-1}(U) \mapsto U \times \mathbb{R}^{k}$$
Now surely, I can take an open set $V \subset R^{k}$ which is not identical with all of $R^{k}$ and take the (open) set (in the product space) $U\times V$ which is mapped to an open subset $W$ of $\pi^{-1}(U)$ because $\Phi$ is a homeomorphism. This set $W$ is open (because $\pi^{-1}(U)$ is open) and its $\Phi$-image does not contain all vectors of $\mathbb{R}^{k}$. Informally speaking: It contains points of the manifold but does not associate all available tangent vectors to every point.
Now if the statement on the coarsest topology were true, it would follow that all open sets of $TM$ are given by $\pi^{-1}(O)$ were $O$ is open in $M$. But such a preimage would need to contain all tangent vectors corresponding to points lying in $O$.
This is nonsense. As you write, the coarsest topology making $\pi$ continuous is given by the set of all $\pi^{-1}(U)$ with $U \subset M$ open. Then certainly no fiber-preserving bijection $\Phi : \pi^{-1}(U) \to U \times \mathbb R^k$ (i.e. satisfying $p_U \circ \Phi = \pi$ with projection map $p_U : \pi^{-1}(U) \to U$) can be a homeomorphism.
Lee's construction associates to each chart $(U,\phi)$ on $M$ a fiber-preserving bijection $\Phi : \pi^{-1}(U) \to U \times \mathbb R^k$. Then you can easily show that $TM$ is given the finest topology such that all
$$\tilde \phi : U \times \mathbb R^k \xrightarrow{\Phi^{-1}}\pi^{-1}(U) \hookrightarrow TM$$
are continuous. In other words, $TM$ receives the final topology with respect to the family of injective functions $\{\tilde \phi\}$.