  The Strong Exponential Hierarchy $SEH$ was shown to collapse to $P^{NExp}$ by
Hemachandra by proving $P^{NExp} = NP^{NExp}$ via a census argument.
Nonetheless, Hemachandra also asked for certificate-based and alternating
Turing machine characterizations of the $SEH$ levels, in the hope that these
might have revealed deeper structural reasons behind the collapse. These open
questions have thus far remained unanswered.
  To close them, by building upon the notion of Hausdorff reductions, we
investigate a natural normal form for the intermediate levels of the
(generalized) exponential hierarchies, i.e., the single-, the
double-Exponential Hierarchy, and so on. Although the two characterizations
asked for derive from our Hausdorff characterization, it is nevertheless from
the latter that a surprising structural reason behind the collapse of $SEH$ is
uncovered as a consequence of a very general result: the intermediate levels of
the exponential hierarchies are precisely characterized by specific "Hausdorff
classes", which define these levels without resorting to oracle machines. By
this, contrarily to oracle classes, which may have different shapes for a same
class (e.g., $P^{NP}_{||} = P^{NP[Log]} = LogSpace^{NP}$), hierarchy
intermediate levels are univocally identified by Hausdorff classes (under the
hypothesis of no hierarchy collapse). In fact, we show that the rather simple
reason behind many equivalences of oracle classes is that they just refer to
different ways of deciding the languages of a same Hausdorff class, and this
happens also for $P^{NExp}$ and $NP^{NExp}$.
  In addition, via Hausdorff classes, we define complete problems for various
intermediate levels of the exponential hierarchies. Through these, we obtain
matching lower-bounds for problems known to be in $P^{NExp[Log]}$, but whose
hardness was left open due to the lack of known $P^{NExp[Log]}$-complete
problems.
