Separation systems are posets with additional structure that form an abstract setting in which tangle-like clusters in graphs, matroids and other combinatorial structures can be expressed and studied.This paper offers some basic theory about infinite separation systems and how they relate to the finite separation systems they induce. They can be used to prove tangle-type duality theorems for infinite graphs and matroids, which will be done in future work that will build on this paper. IntroductionThis paper is a sequel to, and assumes familiarity with, an earlier paper [4] in which finite abstract separation systems were introduced. The profinite separation systems introduced here, and the results proved, will form the basis for our proof of the tangle duality theorem for infinite graphs [1], as well as for a more comprehensive study of profinite tree sets [9].Abstract separation systems were introduced in [4] to lay the foundations for a comprehensive study of how tangles, originally introduced by Robertson and Seymour [12] in the course of their graph minors project, can be generalized to capture, and relate, various other types of highly cohesive regions in graphs, matroids and other combinatorial structures. The basic idea behind tangles is that they describe such a region indirectly: not by specifying which objects, such as vertices or edges, belong to it, but by setting up a system of pointers on the entire structure that point towards this region. The advantage of this indirect approach is that such pointers can locate such a highly cohesive region even when it is a little fuzzy -e.g., when for every low-order separation of a graph or matroid 'most' of the region will lie on one side or the other, so that this separation can be oriented towards it and become a pointer, but each individual vertex or edge (say) can lie on the 'wrong' side of some such separation. (The standard example is that of a large grid in a graph: for every low-oder separation of the graph, most of the grid will lie on one side, but each of its vertices lies on the 'other' side of the small separator that consists of just its four neighbours.)This indirect approach can be applied also to capture fuzzy clusters in settings very different from graphs; see [2,3,6,7,8] for more discussion.Robertson and Seymour [12] proved two major theorems about tangles: the tangle-tree theorem, which shows how the tangles in a graph can be pairwise separated by a small set of nested separations (which organize the graph into a 1 arXiv:1804.01921v3 [math.CO]

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Tree-like decompositions play an important role both in structural combinatorics and algorithm design. The most well-known are [tree decompositions](https://en.wikipedia.org/wiki/Tree_decomposition) of graphs, which are of key importance in the Graph Minor Project of Robertson and Seymour as well as in algorithm design. The minimum width of a tree decomposition is the [treewidth](https://en.wikipedia.org/wiki/Treewidth) of a graph; this parameter captures how closely the global structure of a graph resembles a tree. In particular, a graph has treewidth one if and only if each of its components is a tree. Treewidth has a strong [duality property](https://en.wikipedia.org/wiki/Duality_(optimization)) in the sense of mathematical optimization: the minimum treewidth of a graph is equal to the maximum order of a [bramble](https://en.wikipedia.org/wiki/Bramble_(graph_theory)) increased by one; a bramble is a collection of touching connected subgraphs that cannot be hit by a small set of vertices. [Branchwidth](https://en.wikipedia.org/wiki/Branch-decomposition), which is linearly related to treewidth, is another important width parameter, and its dual parameter is the maximum order of a *tangle*, an object that is more amenable to generalizations to other combinatorial structures than a bramble. Informally speaking, a tangle picks for every separation a small part in a way that small parts of any three separations do not cover the whole structure. In the case of graphs, a tangle of order $k$ concerns vertex $k$-separations, i.e., pairs of subsets of the vertex set such that the union of the two subsets is the whole vertex set, the intersection has size at most $k$ and every edge is in one of the subsets. The main result of the article asserts that every tangle of a graph can be obtained by assigning weights to the vertices so that the small part of every $k$-separation is the one of smaller weight.

We show that every structurally submodular separation system admits a canonical tree set which distinguishes its tangles.

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type ω + 1. Then we introduce and study a topological generalisation of infinite trees which can have limit edges, and show that every infinite tree set can be represented by the tree set admitted by a suitable such tree-like space.