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 $W_1,W_2,W_3,\dots$ 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 $W_1^{\prime },W_2^{\prime },W_3^{\prime },\dots$ that are weakly isomorphic to $W_1,W_2,W_3,\dots$. 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 Lovász and Szegedy about compactness of the space of graphons. We connect these results to “multiway cut” characterization of cut distance convergence from Borgs et al. (2012) [5].

These results are more naturally phrased in the Vietoris hyperspace $K({\mathcal{W}}_0)$ over graphons with the weak* topology. We show that graphons with the cut distance topology are homeomorphic to a closed subset of $K({\mathcal{W}}_0)$, 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 “structuredness order”.

石墨烯理论最终与所谓的切割规范相关。在本文中，我们通过弱*拓扑（当考虑先验的${\mathrm{大号}}^{\mathrm{1个}}$-职能）。我们证明一个序列${\mathrm{w\; ^}}_{\mathrm{1个}}，{\mathrm{w\; ^}}_2，{\mathrm{w\; ^}}_3，\dots$ 当且仅当我们对所有石墨烯序列的弱*累积点集和弱*极限点集相等时，才可在一定距离内收敛石墨烯 ${\mathrm{w\; ^}}_{\mathrm{1个}}^{\prime }，{\mathrm{w\; ^}}_2^{\prime }，{\mathrm{w\; ^}}_3^{\prime }，\dots$ 弱同构的 ${\mathrm{w\; ^}}_{\mathrm{1个}}，{\mathrm{w\; ^}}_2，{\mathrm{w\; ^}}_3，\dots$。我们进一步给出简短的描述性集合理论论点，即每个石墨烯序列都包含具有上述属性的子序列。尤其是，这提供了洛瓦兹（Lovász）和塞格迪（Szegedy）关于石墨烯空间紧凑性定理的另一种证明。我们将这些结果与Borgs等人的切割距离收敛的“多路切割”特征联系起来。（2012）[5]。

这些结果在Vietoris超空间中更自然地表述 $ķ（{\mathcal{w\; ^}}_0）$在具有弱*拓扑的石墨烯上。我们显示了具有割距拓扑的石墨烯对的闭合子集是同胚的$ķ（{\mathcal{w\; ^}}_0）$，并得出此事实的若干后果。

从这些概念出发，出现了关于石墨烯空间的新秩序。此顺序允许比较两个石墨烯的结构。我们建立了这种“结构性秩序”的基本属性。