Relating the cut distance and the weak* topology for graphons

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