# Defining bundles

When introducing tangent spaces on a manifold $${M^{n}}$$, we defined the tangent bundle to be the set of tangent spaces at every point within the region of a coordinate chart $${U\rightarrow\mathbb{R}^{n}}$$, i.e. it was defined as the cartesian product $${U\times\mathbb{R}^{n}}$$. Globally, we had to use an atlas of charts covering $${M}$$, with coordinate transformations $${\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}}$$ defining how to consider a vector field across charts. We now want to take the same approach to define the global version of the tangent bundle, with analogs for frames and internal spaces.