  The theory of graphons is ultimately connected with the so-called cut norm.
In this paper, we approach the cut norm topology via the weak* topology (when
considering a predual of $L^{1}$-functions). We prove that a sequence
$W_1,W_2,W_3,\ldots$ of graphons converges in the cut distance if and only if
we have equality of the sets of weak* accumulation points and of weak* limit
points of all sequences of graphons $W_1',W_2',W_3',\ldots$ that are weakly
isomorphic to $W_1,W_2,W_3,\ldots$. We further give a short descriptive set
theoretic argument that each sequence of graphons contains a subsequence with
the property above. This in particular provides an alternative proof of the
theorem of Lov\'asz and Szegedy about compactness of the space of graphons. We
connect these results to "multiway cut" characterization of cut distance
convergence from [Ann. of Math. (2) 176 (2012), no. 1, 151-219].
  These results are more naturally phrased in the Vietoris hyperspace $K$ over
graphons with the weak* topology. We show that graphons with the cut distance
topology are homeomorphic to a closed subset of $K$, and deduce several
consequences of this fact.
  From these concepts a new order on the space of graphons emerges. This order
allows to compare how structured two graphons are. We establish basic
properties of this "structurdness order".
