Given a Riemannian manifold $M$, What is the topology of its tangent space?
There are two interpretations:
1). View each $T_pM$ as a subspace of the tangent bundle, then its topology should be the subspace topology.
2). Since $M$ is a Riemannian manifold, $T_pM$ is a normed vector space. Then it has a canonical smooth structure as a normed vector space.
I think the second interpretation is more reasonable. Since it is well know that the exponential map $exp$ when restricted to $T_pM$ is a local diffeomorphism around zero. To talk about local diffeomorphism, apparently $T_pM$ should have a smooth structure.
$T_pM$ is a finite-dimensional normed vector space, so the topology is that of $\mathbb{R}^d$, where $d = \mathrm{dim}\; M$.