The base manifold in a bundle is analogous to the domain for a set of functions. In fact, a bundle, by definition, comes with a map to the base manifold, often called or projection.
For example, the base manifold to the tangent bundle of a manifold is the manifold . A vector field is a function from the manifold to the tangent bundle, with the restriction that every point gets mapped to a vector at that point. In general, a bundle has bundle sections, at least locally, which are maps from the base manifold to the bundle.