I've been reading the wiki page https://en.wikipedia.org/wiki/Tangent_bundle of the tangent bundle and it states that the tangent bundle is equipped with the natural topology and not the disjoint union topology. Nothing further is explained.

Can someone explain to me the ramifications of these topologies?

The disjoint union topology would make each tangent space $T_x M$ a connected component; so (unless $M$ is zero-dimensional) you get uncountably many connected components, which contradicts the requirement of second-countability in the usual definition of a manifold.

More importantly, though, this topology doesn't give us a notion of two tangent vectors based at nearby (but distinct) points being close, so it's not useful for doing the kind of things we want to do with tangent vectors. For example, the velocity vector field $[0,1] \to TM$ of a smooth curve $[0,1] \to M$ is continuous if we put the natural topology on $TM$, but not with the disjoint union topology.

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