  Bundles of matrix polynomials are sets of matrix polynomials with the same
size and grade and the same eigenstructure up to the specific values of the
eigenvalues. It is known that the closure of the bundle of a pencil $L$
(namely, a matrix polynomial of grade $1$), denoted by $\mathcal{B}(L)$, is the
union of $\mathcal{B}(L)$ itself with a finite number of other bundles. The
first main contribution of this paper is to prove that the dimension of each of
these bundles is strictly smaller than the dimension of $\mathcal{B}(L)$. The
second main contribution is to prove that also the closure of the bundle of a
matrix polynomial of grade larger than 1 is the union of the bundle itself with
a finite number of other bundles of smaller dimension. To get these results we
obtain a formula for the (co)dimension of the bundle of a matrix pencil in
terms of the Weyr characteristics of the partial multiplicities of the
eigenvalues and of the (left and right) minimal indices, and we provide a
characterization for the inclusion relationship between the closures of two
bundles of matrix polynomials of the same size and grade.
