So I'm trying to learn about fibre bundles and I am looking at the example of a tangent bundle. Given a differentiable manifold M. Denote the tangent space at [tex]p \in M[/tex] by [tex]T_p M[/tex]. Is the definition of the tangent bundle [tex]TM = \lbrace (p, T_p M)|p \in M \rbrace[/tex] or is it [tex]TM = \lbrace (p, V)|p \in M , V \in T_p M\rbrace[/tex]? Maybe I'm splitting hairs but there should be standard definition of one or the other, right? I can discuss further why I think it matters but first let's just see if anyone is certain about the answer.