  There is a rich theory of maximal order types of well-partial-orders (wpos),
pioneered by de Jongh and Parikh (1977) and Schmidt (1981). Every wpo is
Noetherian in its Alexandroff topology, and there are more; this prompts us to
investigate an analogue of that theory in the wider context of Noetherian
spaces.
  The notion of maximal order type does not seem to have a direct analogue in
Noetherian spaces per se, but the equivalent notion of stature, investigated by
Blass and Gurevich (2008) does: we define the stature $||X||$ of a Noetherian
space $X$ as the ordinal rank of its poset of proper closed subsets. We obtain
formulas for statures of sums, of products, of the space of words on a space
$X$, of the space of finite multisets on $X$, in particular. They confirm
previously known formulas on wpos, and extend them to Noetherian spaces.
  The proofs are, by necessity, rather different from their wpo counterparts,
and rely on explicit characterizations of the sobrifications of the
corresponding spaces, as obtained by Finkel and the first author (2020).
  We also give formulas for the statures of some natural Noetherian spaces that
do not arise from wpos: spaces with the cofinite topology, Hoare powerspaces,
powersets, and spaces of words on $X$ with the so-called prefix topology.
  Finally, because our proofs require it, and also because of its independent
interest, we give formulas for the ordinal ranks of the sobrifications of each
of those spaces, which we call their sobrification ranks.
