Is the topology on the tangent bundle initial wrt projection map to the base manifold?

Question

Given a manifold $M$, and the tangent bundle over it: $\pi:TM\rightarrow M$; is it correct to say that the topology on $TM$ is the intial one induced by the projection map $\pi$?

Answer 1

By no means. It's initial for the trivializations, see here. The initial topology for the projection is extremely coarse: the basis is the inverse images $U\times \mathbb{R}^n$ of open subsets $U\subset M$. This is just the product $M\times \mathbb{R}^n,$ but where $\mathbb{R}^n$ is given the indiscrete topology.