Different Topologies for a Tangent Bundle $TM$ of a smooth manifold $M$

Question

I am following the comment of Dylan Moreland for this thread: Topology of the tangent bundle of a smooth manifold.

I was trying to understand, why putting a disjoint union topology on $TM=\sqcup_{p\in M}T_pM$ is not a good idea? In what way different tangents spaces are not going to behave nicely?