  We show that a diffeological bundle gives rise to an exact sequence of
internal tangent spaces. We then introduce two new classes of diffeological
spaces, which we call weakly filtered and filtered diffeological spaces, whose
tangent spaces are easier to understand. These are the diffeological spaces
whose categories of pointed plots are (weakly) filtered. We extend the exact
sequence one step further in the case of a diffeological bundle with filtered
total space and base space. We also show that the tangent bundle $T^H X$
defined by Hector is a diffeological vector space over $X$ when $X$ is filtered
or when $X$ is a homogeneous space, and therefore agrees with the dvs tangent
bundle introduced by the authors in a previous paper.
