Representations of infinite tree-sets

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 ${\omega+1}$. 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.


§1. Introduction
Separations of graphs have been studied in the context of structural graph theory for a long time. For instance every edge of the decomposition tree of a tree-decomposition of a graph defines a separation in a natural way * . The separations obtained in this way have an additional important property: they are nested † with each other. Looking at nested sets of separations of a graph has since been a useful way to study tree-decompositions, and especially in infinite graphs they offer an analogue when a tree-decomposition with a certain desired property may not exist (see [8] for example).
While any tree-decomposition of a graph into small parts witnesses that the graph has low tree-width, there are various dense objects that force high tree-width in a graph.
Among these are large cliques and clique minors, large grids and grid minors as well as high-order brambles. All these dense objects in a graph have the property that they orient its low-order separations by lying mostly on one side of any given low-order separation.
For such a dense structure in a graph these orientations of separations are consistent with each other: no two of them 'disagree' about where the dense object lies by pointing away from each other.
In [7] Robertson and Seymour proposed the notion of tangles, which are such families of consistently oriented separations up to a certain order. These tangles can be studied in their own right, instead of any dense objects that may induce them. By varying the * As the sides of the separation, consider the union of the parts corresponding to the components of the tree after deleting the edge.
† Two separations are nested if a side of one separation is a subset of a side of the other, and vice versa.
strength of the consistency conditions one can model different kinds of dense objects, and the resulting consistent orientations give rise to different types of tangles.
To talk about these separations systems one does not even need an underlying graph structure or ground set: they can be formulated in a purely axiomatic way, see Diestel [2]. Such a separation system is simply a partially ordered set with an order-reversing involution.
The notions of consistency of separations that come from dense substructures in graphs can be translated into this setting as well. The tangles of graphs then become abstract tangles, and the tree-like structures become nested systems of separations, so-called tree sets [3]. This abstract framework turns out to be no less powerful, even for graphs alone, than ordinary graph separations. In [4] Diestel and Oum established an abstract duality theorem for separation systems which easily implies (see [5]) all the classical duality results from graph-and matroid theory, such as the tree-width duality theorem by Seymour and Thomas [9]. The unified duality theorem asserts that for any sensible notion of consistency a separation system contains either an abstract tangle or a tree set witnessing that no such tangle exists. Furthermore this abstract notion of separation systems can be applied in fields outside of graph theory, for instance in image analysis [6].
Tree sets are also interesting objects in their own right: they are flexible enough to model a whole range of other 'tree-like' structures in discrete mathematics, such as ordinary graph trees, order trees and nested systems of bipartitions of sets [3].
In fact, tree sets and graph-theoretic trees are related even more closely than that: for any tree T the set Ñ E of oriented edges of T admits a natural partial order, which in fact turns Ñ E into a tree set, the edge tree set of T . As was shown in [3], these edge tree sets of graph-theoretical trees are rich enough to represent all finite tree sets: every finite tree set is isomorphic to the edge tree set of a suitable tree.
In this paper we extend the analysis of representations of tree sets to infinite tree sets.
The definition of an edge tree set of a graph-theoretical tree straightforwardly extends to infinite trees. From the structure of these it is clear that the edge tree set of a tree T cannot contain a chain of order type ω`1. We will show that this is the only obstruction for a tree set to being representable by the edge tree set of a (possibly infinite) tree: Theorem 1. Every tree set without a chain of order type ω`1 is isomorphic to the edge tree set of a suitable tree.
Secondly, we would like to represent infinite tree sets that do contain a chain of order type ω`1 by edge tree sets of an adequate tree structure as well. To achieve this we turn to the notion of graph-like spaces introduced by Thomassen and Vella [10] and further studied by Bowler, Carmesin and Christian [1]: these are topological spaces with a clearly defined structure of vertices and edges, which can be seen as a limit object of finite graphs.
In particular, for a chain of any order type, there exists a graph-like space containing a 'path' whose edges form a chain of that order type. Therefore the tree-like spaces, those graph-like spaces which have a tree-like structure, overcome the obstacle of chains of order type ω`1 which prevented the edge tree sets of infinite trees from representing all infinite tree sets: unlike graph-theoretic trees, tree-like spaces can have limit edges. And indeed we will prove in this paper that the edge tree sets of tree-like spaces can be used to represent all tree sets.

Theorem 2.
Every tree set is isomorphic to the edge tree set of a suitable tree-like space.
This paper is organised as follows. In Section 2 we recall the basic definitions of abstract separation systems and tree sets and establish a couple of elementary lemmas we will use throughout the paper. Following that, in Section 3, we formally define the edge tree set of a tree and prove Theorem 1. In Section 4, we introduce the concept of tree-like spaces which generalise infinite graph-theoretical trees. We define edge tree sets of tree-like spaces analogously to edge tree sets of graph-theoretical trees and then prove Theorem 2. In order to do this we need a result linking the two concepts of connectivity in graph-like spaces: topological connectivity and 'pseudo-arc connectivity', the analogue of graph-theoretical connectivity for graph-like spaces. In Section 4 we make use of the fact that for compact graph-like spaces these two notion of connectivity are equivalent, and give a proof of this A tree set is a nested separation system with no trivial elements. It is regular if all of its elements are regular, i.e. if no Ñ s P τ is small.
An orientation of a set For a tree set τ an orientation O of τ is splitting if it is consistent and has the property Consistent orientations of a tree set τ can be thought of as the 'vertices' of a tree set, an idea that we will make more precise in the next sections. In the context of infinite tree sets, the non-splitting orientations can be thought of as 'limit vertices' or 'ends' of the tree set. More generally, given a partial orientation P of τ , is it possible to extend it to a consistent orientation of τ ? Of course P needs to be consistent itself for this to be possible. The next Lemma shows that under this necessary assumption it is always possible to extend a partial orientation to all of τ . In particular, every element of a tree set induces a consistent orientation in which it is a maximal element. This orientation is in fact unique: where Ñ s i ď Ñ s j and Ð s i ě Ð s j whenever i ď j, as well as Ñ s n ď Ñ t and Ð s n ě Ð t for all n P N.
The separation Ð t is maximal in the orientation which is not splitting as no Ñ s n lies below a maximal element of O. Hence Ð t does not lie in a splitting star of τ .
In the above example the chain C " t Ñ s n | n P Nu Y t Ñ t u has order-type ω`1. But these ω`1 chains turn out to be the only obstruction for separations not being elements of splitting starts, as the following lemma shows. Let us call a tree set that does not contain a chain of order type ω`1 tame. A direct consequence of Lemma 2.4 is that every element of a finite tree set lies in a splitting star.
Given two separation systems R and S, a map f : R Ñ S is a homomorphism of separation systems if it commutes with the involution, i.e. pf p Ñ r qq˚" f p Ð rq for all Ñ r P R, and is order- Please note that the condition for f to be order-preserving is not 'if and only if': it is allowed that f p Ñ r 1 q ď f p Ñ r 2 q for incomparable Ñ r 1 , Ñ r 2 P R. Furthermore f need not be injective.
As all trivial separations are small every regular nested separation system is a tree set.
These two properties, regular and nested, are preserved by homomorphisms of separations systems, albeit in different directions: the image of nested separations is nested, and the preimage of regular separations is regular. Proof. First suppose that some Ñ r P R is small, that is, that Ñ r ď Ð r. Then f p Ñ r q ď f p Ð rq " pf p Ñ r qq˚, so S contains a small element. Therefore if S is regular then R must be regular as well.
Now suppose that R is nested consider two unoriented separations s, s 1 P S and for which there are r, r 1 P R with s " f prq and s 1 " f pr 1 q. Since R is nested r and r 1 have comparable Proof. From Lemma 2.5 it follows that both R and S are regular and nested, which means they are regular tree sets. Therefore all we need to show is that the inverse of f is order-preserving, i.e. that Ñ r 1 ď Ñ r 2 whenever f p Ñ r 1 q ď f p Ñ r 2 q. Let Ñ r 1 , Ñ r 2 P R with f p Ñ r 1 q ď f p Ñ r 2 q be given. As R is nested, r 1 and r 2 have comparable orientations.
If Ñ r 1 ě Ñ r 2 , then f p Ñ r 1 q " f p Ñ r 2 q, implying Ñ r 1 " Ñ r 2 and hence the claim. If Ñ r 1 ď Ð r 2 , then f p Ñ r 1 q ď f p Ñ r 2 q, f p Ð r 2 q, contradicting the fact that S is a regular tree set. Finally, as desired. §3. Regular tame tree sets and graph-theoretical trees Every graph-theoretical tree T naturally gives rise to a tree set, its edge tree set τ pT q of T (see below for a formal definition). However, while every tree gives rise to a tree set, not every tree set 'comes from' a tree. In this section we characterise those infinite tree sets that arise from graph-theoretical trees as the tree sets which are both regular and tame, i.e. contain no chain of order-type ω`1. More precisely, given a regular tame tree set τ we will define a corresponding tree T pτ q. These definitions in turn should be able to capture the essence of what it means to be 'tree-like'. More precisely we want the following properties: ‚ the tree constructed from the edge tree set of T is isomorphic to T ; ‚ the edge tree set of the tree constructed from τ is isomorphic to τ .
3.1. The edge tree set of a tree. Let T " pV, Eq be a graph-theoretical tree, finite or infinite. Let Ñ EpT q be the set of oriented edges of T , that is We define an involution˚by setting px, yq˚:" py, xq for all edges xy P EpT q, and a partial order ď on Ñ EpT q by setting px, yq ă pv, wq for edges xy, vw P EpT q if and only if tx, yu ‰ tv, wu and the unique tx, yu-tv, wu-path in T joins y to v. Then the edge tree set τ pT q is the separation system p Ñ EpT q, ď,˚q. It is straightforward to check that τ pT q is indeed a regular tree set.
Note that every maximal chain in τ pT q corresponds to the edge set of a path, ray or double ray in T . Hence τ pT q does not contain any chain of length ω`1 and hence is tame.
If T is the decomposition tree of a tree-decomposition of a graph G, then the tree set τ pT q is isomorphic to the tree set formed by the separations of G that correspond § to the edges of T (with some pathological exceptions). This relationship between tree-decompositions and tree sets was further explored in [3].
3.2. The tree of a regular tame tree set. Let τ be a regular tame tree set. Our aim is to construct a corresponding graph-theoretical tree T pτ q. Recall that a consistent We will use O as the vertex set of T pτ q. Moreover note that it will turn out that the non-splitting orientations will precisely correspond to the ends of T pτ q.
Let us show first that, for any two splitting stars, each of them contains exactly one element that is inconsistent with the other star. We will later use this little fact when we define the edges of our tree.
Proof. There is at least one such Ñ s as O 2 does not induce σ 1 . For any two Ñ r , Ñ s P σ the Note that this lemma holds for every tree set as the proof did not use any assumptions on τ .
Our assumption that τ is tame implies the following sufficient condition for a consistent orientation to be splitting: Let O be a consistent orientation of τ with at least one maximal element.
Then O splits τ .
Proof. Let Together with the Extension Lemma 2.2 this immediately implies the following: Proof. For Ñ s P τ apply the Extension Lemma 2.2 to t Ñ s u to obtain a unique consistent orientation O of τ in which Ñ s is a maximal element. It then follows from Lemma 3.2 that O is splitting.
For Ñ s P τ write Op Ñ s q for the unique consistent orientation of τ in which Ñ s is maximal.
Now we define the graph T pτ q. Let V pT pτ qq " O and We call T pτ q the tree corresponding to τ , where τ is a regular tame tree set.
First note that T pτ q does not contain any loops and hence is indeed a simple graph since Op Ñ s q and Op Ð s q are different for any Ñ s P τ .
We need to check that T pτ q is a tree. To prove that T pτ q is connected, our strategy is as follows. Either this process terminates after finitely many steps, in which case we found a path from O to O 1 , or it continues indefinitely. In the latter case the infinitely many separations we inverted form a chain with an upper bound in O 1 , which would yield a chain of order type ω`1.
The next short Lemma forms the basis of this iterative flipping process.
Proof. Let Ñ s n`1 be the unique separation in Op Ñ s 1 q with Op Ð s n`1 q " Op Ñ s n q. Then Ñ s n ď Ñ s n`1 by the star property. Hence if Ñ s n`1 ď Ð s 1 , then Ñ s n would be trivial, therefore Ñ s n`1 ď Ñ s 1 as desired.
For Ñ s 1 , . . . , Ñ s n , Ñ s 1 and Ñ s n`1 as in Lemma 3.5 there is an edge between Op Ñ s k q and Op Ð s k`1 q for every 1 ď k ď n. Additionally if Ñ s n`1 ‰ Ñ s 1 then Ñ s 1 , . . . , Ñ s n`1 , Ñ s 1 again fulfill the assumptions of the lemma, so it can be used iteratively.
Furthermore note that Ñ s 1 ď Ñ s 2 ď¨¨¨ď Ñ s n ď Ñ s n`1 , so if this iteration does not terminate the Ñ s k form an infinite chain. From this we now prove that T pτ q is connected.
Otherwise the assumptions of Lemma 3.5 are met for n " 1. Applying Lemma 3.5 iteratively either yields Ñ s n`1 " Ñ s 1 for some n P N, in which case we found a path in T pτ q joining O and O 1 , or we obtain a strictly increasing sequence p Ñ s n q nPN with Ñ s n ď Ñ s 1 for all n P N, that is, a chain of order type ω`1.

Regular tame tree sets and trees -A characterisation.
Finally we will prove that the given constructions of the previous subsections agree with each other.
Proof. Let ϕ : τ 1 Ñ τ pT pτ 1 qq be the map defined by ϕp Ñ s q " pOp Ð s q, Op Ñ s qq. This is a bijection by Corollary 3.3. Note that for Ñ s P τ 1 the orientations Op Ð s q and Op Ñ s q differ only in s by consistency and are thus adjacent in T .
As τ 1 and τ pT pτ 1 qq are regular tree sets all we need to show is that ϕ is a homomorphism of separation systems. Then ϕ will be an isomorphism of tree sets by Lemma 2.6.
It is clear from the definition that ϕ commutes with the involution. Therefore it suffices to show that ϕ is order-preserving.
Let Ñ s , Ñ s 1 P τ 1 be two separations with Ñ s ă Ñ s 1 . We need to show that the unique whose nodes contains Ñ s and Ð s 1 by consistency. Hence ϕp Ñ s q ă ϕp Ñ s 1 q as desired.
Lemma 3.8. Any graph-theoretic tree T 1 is isomorphic to T pτ pT 1 qq.
Otherwise, for each node v P V pT 1 q there is at some oriented edge pw, vq P EpT 1 q pointing towards that node. Let ϕ : T 1 Ñ T pτ pT 1 qq be the map defined by ϕpvq :" Oppw, vqq. This map is well-defined since the edges directed towards a node v P V pT 1 q form a splitting star with the same maximal elements yielding the unique consistent orientation containing all these oriented edges (cf. Corollary 3.3).
Similarly, given some O " Oppw, vqq P V pT pτ pT 1 qqq, we obtain ϕpvq " O and hence that ϕ is surjective. By construction there is an edge between Oppv, wqq and Oppw, vqq for any edge vw P EpT q and similarly no edge between Oppv, wqq and O if pw, vq is not Hence we have proven our main theorem of this section: (1) A tree set is isomorphic to the edge tree set of a tree if and only if it is regular and tame.
Additionally, for distinct but comparable tree sets, we can say precisely in which way the corresponding trees from Theorem 3.9 above are comparable: one will be a minor of the other.
(2) If T 1 is a minor of T 2 , then τ pT 1 q is isomorphic to a subset of τ pT 2 q.
Theorem 3.10 is a special case of Theorems 4.14 and 4.15 from the next section and hence we will omit its proof here. §4. Regular tree sets and tree-like spaces 4.1. Graph-like spaces. As we have seen in Section 3, not every tree set, even regular, can be represented as the edge tree set of a tree. In this section we find a (topological) relaxation of the notion of a (graph-theoretical) tree, to be called tree-like spaces. Like trees, these tree-like spaces give rise to a regular edge tree set in a natural way, but which are just general enough that, conversely, every regular tree set can be represented as the edge tree set of a tree-like space.
The concept of graph-like spaces was first introduced in [10] by Thomassen and Vella, and further studied in [1] by Bowler, Carmesin and Christian. In [1] the authors discuss the connections between graph-like spaces and graphic matroids, which are of no interest to us here. Instead we determine when a graph-like space is tree-like, and then show that every regular tree set can be represented as the edge tree set of a tree-like space.
Graph-like spaces are limit objects of graphs that are not themselves graphs. In short they consist of the usual vertices and edges, together with a topology that allows the vertices and edges to be limits of each other. The formal definition is as follows.
together with a vertex set V pGq, an edge set EpGq and for each e P EpGq a continuous map ι G e : r0, 1s Ñ G (the superscript may be omitted if G is clear from the context) such that: ‚ The underlying set of G is V pGq 9 Yrp0, 1qˆEpGqs.
‚ For any x P p0, 1q and e P EpGq we have ι e pxq " px, eq.
‚ ι e p0q and ι e p1q are vertices (called the end-vertices of e).
‚ For any two distinct v, v 1 P V pGq, there are disjoint open subsets U, U 1 of G partitioning V pGq and with v P U and v 1 P U 1 .
The inner points of the edge e are the elements of p0, 1qˆteu.
Note that G is always Hausdorff. For an edge e P EpGq the definition of graph-like space allows ι e p0q " ι e p1q. We call such an edge a loop. In our discussions of graph-like spaces loops are irrelevant, so the reader may imagine all graph-like spaces to be loop-free.
If U and U 1 are disjoint open subsets of G partitioning V pGq we call the set of edges with end-vertices in both U and U 1 a topological cut of G and say that the pair pU, U 1 q induces that cut. The last property of graph-like spaces then says that any two vertices can be separated by a topological cut.
A graph-like space G 1 is a sub-graph-like space of a graph-like space G if V pG 1 q Ď V pGq, EpG 1 q Ď EpGq and G 1 is a subspace of G (as topological spaces). By slight abuse of notation we will write G 1 Ď G to say that G 1 is a sub-graph-like space of G.
Let G be a graph-like space and F Ď EpGq a set of edges of G. We write G´F for the sub-graph-like space G tpx, eq | x P p0, 1q, e P F u with the same vertex set as G, with edge set EpGq F and ι G´F e " ι G e for all e P EpGq F . We abbreviate G´teu as G´e. Given a set W Ď V pGq of non-end-vertices we write G´W for the sub-graph-like space G W with V pG´W q :" V pGq W , EpG´W q :" EpGq and ι G´F e " ι G e for all e P EpGq. For reasons of cardinality arc-connectedness is not a very useful notion in graph-like spaces. Instead we work with an adapted concept of arcs. A graph-like space P is a pseudo-arc if P is a compact connected graph-like space with a start-vertex a and an end-vertex b satisfying the following: ‚ for each e P EpP q the vertices a and b are separated in P´e; ‚ for any two x, y P V pP q there is an edge e P E such that x and y are separated in P´e.
If P contains an edge then a ‰ b; otherwise we call P trivial. A graph-like space G is pseudo-arc-connected if for all vertices a, b P V pGq there is a pseudo-arc P Ď G with start-vertex a and end-vertex b.
The adapted notion of circles is analogous. A graph-like space is a pseudo-circle if it is a compact connected graph-like space with at least one edge satisfying the following: ‚ removing any edge from C does not disconnect C but removing any pair does; ‚ any two vertices of C can be separated in C by removing a pair of edges.
Pseudo-arcs and pseudo-circles are related as follows: Let G be a graph-like space, C a pseudo-circle in G and e P EpCq. Then C´e is a pseudo-arc in G joining the end-vertices of e.
Conversely, let P and Q be nontrivial non-loop pseudo-arcs in G that meet precisely in their end-vertices. Then P Y Q is a pseudo-circle in G.
Given two graph-like spaces G 1 , G 2 , a map ϕ : G 1 Ñ G 2 is an isomorphim of graph-like spaces if it is a homeomorphism (for the topological spaces) and it induces a bijection between V pG 1 q and V pG 2 q.
Let G be a graph-like space and F Ď EpGq a set of edges of G. We define an relation " 1 F on G via ι e pxq " F ι e pyq for all e P F and x, y P r0, 1s.
Let " F denote the minimal equivalence relation that extends the transitive and reflexive closure of " 1 F such that the resulting quotient space G{F :" G{ " F is Hausdorff.

Remark 4.3.
The contraction G{F of F in G is a graph-like space with vertex set V pG{F q :" trvs P G{ " F | v P V pGqu, edge set ¶ EpG{F q :" EpGq F and for each edge One can also easily show that each equivalence class with respect to " F is connected in G. Moreover, we write G.F for G{pEpGq F q for the contraction to F in G.
We say that a graph-like space G 1 is a minor of graph-like space G if there are disjoint edge sets F 1 , F 2 Ď EpGq and a set W Ď V pG{F 1 q´F 2 q of non-end-vertices such that G 1 is isomorphic to ppG{F 1 q´F 2 q´W .
We will also need the following fact about graph-like spaces: ¶ This is a slight abuse of notation since technically the inner points of an edge e in the quotient space are of the form tpx, equ and not px, eq.

Theorem 4.4. A compact graph-like space is connected if and only if it is pseudo-arc connected.
As the proof of Theorem 4.4 is relatively long and does not involve any tree-like spaces or other tree structures, we shall use Theorem 4.4 in this section without proof. Section 5 will then be devoted entirely to proving Theorem 4.4.

Tree-like spaces.
There are many different equivalent ways of defining the graphtheoretical trees, which is an easy exercise to prove. (ii) T is connected but T´e is not for any edge e P EpT q; (iii) T is connected and contains no cycle.
(iv) T contains no cycle but every graph T 1 with V pT 1 q " V pT q and T 1´F " T for some non-empty F Ď EpT 1 q EpT q does.
A graph T is a tree if it has one (and thus all) of the above properties. In some situations one of these properties is easier to work with than the others, and their equivalence is used implicitly in many places in graph theory.
The above properties can be translated into the setting of graph-like spaces to say when a graph-like space is tree-like as follows: (ii) G is connected but G´e is not for any edge e P EpGq; (iii) G is connected and contains no pseudo-circle; (iv) G contains no pseudo-circle but every graph-like space G 1 with V pG 1 q " V pGq and G 1´F " G for some non-empty F Ď EpG 1 q EpGq does.
Analogous to Proposition 4.5, we prove the following proposition. The argument is very similar to the proof of Proposition 4.5, but one additional technical lemma is needed: if two vertices a and b of a graph G are joined by two different paths it is obvious that some edge e P EpGq lies on exactly one of the two paths. However for graph-like spaces and pseudo-arcs this intuitive fact requires a surprising amount of set-up to prove (see [1]).
We forego this technical set-up and simply use the following lemma:   implies that G is pseudo-arc connected. For the uniqueness suppose G contains two different pseudo-arcs P and Q between two vertices a and b. Lemma 4.8 implies that there is an edge e P EpGq which lies on exactly one of the two pseudo-arcs. But then G´e is still pseudo-arc connected and therefore connected, a contradiction.
Similarly to graph-theoretical trees every tree-like space gives rise to a regular tree set, see Subsection 4.3. We will show that the tree-like spaces are rich enough that one can obtain every regular tree set from them. This is in contrast to Section 3 where we showed that the regular tree sets coming from trees are precisely those with no chain of order type ω`1. This restriction was owed to the fact that graph-theoretical trees cannot have edges that are the limit of other edges. But tree-like spaces can have limit edges, so this is no longer a restriction.
In Subsection 4.4 we construct a corresponding regular tree set for a given tree-like space, and in Subsection 4.5 we will prove the characterisation analogously to the one in Section 3 by showing: ‚ the tree-like space constructed from the edge tree set of a tree like space T is isomorphic to T ; ‚ the edge tree set of the tree-like space constructed from a regular tree set τ is isomorphic to τ .

The edge tree set of a tree-like space.
For a tree-like space T we can define the edge tree set τ pT q in a way that is very similar to the definition of τ pT q in Section 3. Let Ñ EpT q :" pι e p0q, ι e p1qqˇˇe P EpT q ( Y pι e p1q, ι e p0qqˇˇe P EpT q ( be the set of oriented edges of T . As tree-like spaces cannot contain loops every element of Ñ EpT q is a pair of two distinct vertices of T . For vertices u, v P V pT q let P pu, vq be the unique pseudo-arc in T with end-vertices u and v. Then τ pT q :" p Ñ EpT q, ď,˚q becomes a separation system by setting px, yq˚:" py, xq and px, yq ă pv, wq for px, yq, pv, wq P Ñ EpT q with tx, yu ‰ tv, wu whenever P py, vq Ď P px, vq Ď P px, wq.
It is straightforward to check that τ pT q is a regular tree set.

4.4.
The tree-like space of a tree set. Let τ " p E, ď,˚q be a regular tree set; we define the tree-like space corresponding to τ , denoted T pτ q. Let V :" Opτ q be the set of consistent orientations and E the set of unoriented separations of τ . As in Section 3 let Op Ñ s q be the unique O P Opτ q in which Ñ s is maximal. We define the tree-like space T pτ q with vertex set V and edge set E, that is with ground set V Y`p0, 1qˆE˘. For this we need to define the maps ι e : r0, 1s Ñ T pτ q.
Fix any orientation O 1 of τ . For each Ñ e P O 1 let ι e : r0, 1s Ñ T be the map So far the definition of V and the adjacencies in T pτ q have been analogous to the construction from Section 3. But to make T pτ q into a graph-like space we also need to define a topology.
For Ñ e P O 1 let E`p Ñ eq be the set of all Ñ s P O 1 with Ñ e ă Ñ s or Ñ e ă Ð s , and E´p Ñ eq the set of For Ñ e P O 1 and r P p0, 1q set We define the sub-base of the topology on T pτ q as S :" Sp Ñ e, rqˇˇÑ e P τ, r P p0, 1q ( . Note that only the notation depends on the choice of O 1 but the topology on T pτ q does not.
It is clear that T pτ q is a graph-like space: for any two vertices a, b P V pick any Ñ e in the symmetric difference of a and b, viewed as orientations of τ . Then Sp Ñ e, 1 2 q and Sp Ð e, 1 2 q are disjoint open sets partitioning V and ta, bu. Lemma 4.9. T pτ q is compact.
Proof. By the Alexander sub-base theorem from general topology it suffices to show that any open covering of sets in S has a finite sub-cover. Suppose that C is a sub-basic open cover of T pτ q with no finite sub-cover. Let EpCq be the set of all Ñ e P τ such that Sp Ñ e, xq P C for some x P p0, 1q. If Ñ r ď Ð s for any Ñ r , Ñ s P EpCq then their corresponding sets in C already cover all of T pτ q, except possibly for p0, 1qˆr if Ñ r " Ð s , which can be finitely covered.
Thus we may assume that Ñ r ď Ð s for all Ñ r , Ñ s P EpCq. Then the set E˚pCq :" t Ð e | Ñ e P EpCqu is a consistent partial orientation of τ , so by the Extension Lemma 2.2 there is an O P Opτ q with E˚pCq Ď O. But O R Sp Ñ e, rq for every Ñ e P EpCq and r P p0, 1q, so C was not a cover of T . Therefore T is a compact graph-like space. any edge e P E the image of ι e in T is connected, hence every edge whose inner points meet A is completely contained in A, and similarly for B. Write τ A for the set of Ñ e P τ withe Ď A, and τ B for the set of Ñ e P τ withe Ď B. Then τ A and τ B partition τ and are closed under involution. Fix any Ñ a P τ A and for the chain of elements between Ñ a and say. Let X Ď τ be minimal in size with the property that for suitable rp Ñ xq P p0, 1q. From our assumptions it follows that such an X exists and is a finite subset of O, and the minimality implies that X is a star. Observe thatb Ď Sp Ñ x, rp Ñ xqq Therefore T pτ q is connected.
Hence we have shown that T pτ q is indeed a tree-like space.

Regular tree sets and tree-like spaces -A characterisation.
Lemma 4.11. Any regular tree set τ 1 is isomorphic to τ pT pτ 1 qq.
Proof. For two vertices u, v P Opτ 1 q the set C " v u is a chain in τ 1 . Set P pu, vq :" ď te | Ñ e P Cu Ď T pτ 1 q.
This is a bijection between τ 1 and Ñ EpT pτ 1 qq that commutes with the involution. The claim follows from Lemma 2.6 if we can show that ϕ is order-preserving. For this let Ñ r , Ñ s P τ 1 * * This follows immediately if one uses the machinery established in [1], which we do not introduce here.
Alternatively one can show the connectedness of P pu, vq by repeating the proof that T pτ 1 q is connected, and verifying the other properties of a pseudo-arc directly.
with Ñ r ă Ñ s . Let px, yq be the end-vertices of r P EpT pτ 1 qq with Ñ r P y and pv, wq the end-vertices of s P EpT pτ 1 qq with Ñ s P w. Then v y " pv xq t Ñ r u and v x " pw xq t Ñ s u, so P py, vq Ď P px, vq Ď P px, wq and hence ϕp Ñ r q " px, yq ď pv, wq " ϕp Ñ s q.
Lemma 4.12. Any tree-like space T 1 is isomorphic to T pτ pT 1 qq.
Proof. For ease of notation, we may assume that without loss of generality that the arbitrary orientation of τ pT 1 q we fixed for the construction of T pτ pT 1 qq is tpι T 1 e p0q, ι T 1 e p1qq | e P EpT 1 qu. For every edge e P EpT 1 q there is a unique jpv, eq P t0, 1u such that v is in the same component of T 1´e as ι T 1 e pjpv, eqq by Proposition 4.7. We define a map ϕ : V pT 1 q Ñ V pT pτ pT 1 qqq by setting ϕpvq to be the orientation tpι T 1 e p1´jpv, eqq, ι T 1 e pjpv, eqqq | e P EpT qu of τ pT 1 q, which is easily verified to be consistent.
We extend ϕ to a map T 1 Ñ T pτ pT 1 qq by setting ϕpr, eq :" pr, tι T 1 e p0q, ι T 1 e p1quq for r P p0, 1q and e P EpT 1 q. It is easy to check that ϕ is a bijection and induces a bijection between V pT 1 q and V pT pτ pT 1 qqq. Since T 1 is compact and T pτ pT 1 qq is Hausdorff, we only need to check that ϕ is continuous. For each e P EpT 1 q and each r P p0, 1q note that T 1 tru contains two connected components Cpe, r, 0q and Cpe, r, 1q, where Cpe, r, jq denotes the component containing ι T 1 e pjq. By construction, ϕpCpe, r, jqq " Sppι T 1 e p1´jq, ι T 1 e pjqq, rq and hence the preimage of any subbasis element is open.
Altogether we have proven the main theorem of this section.

1) A tree set is isomorphic to the edge tree set of a tree-like space if
and only if it is regular.
(3) Any tree-like space T 1 is isomorphic to T pτ pT 1 qq.
Additionally, for distinct but comparable tree sets, we can say precisely in which way the corresponding trees from Theorem 3.9 above are comparable: one will be a minor of the other.
Let us finish this section with two further results on how these constructions relate to substructures.
First we note that Opτ 1 q " tO X τ 1 | O P Opτ 2 qu. Moreover it immediately follows from the definitions that O, O 1 P Opτ 2 q are representatives of the same vertex of T 2 if and only if O X τ 1 " O 1 X τ 1 .
For ease of notation we may assume without loss of generality that the orientation of τ 1 that we chose in the construction of T pτ 1 q is induced by the orientation we chose for τ 2 in the construction of T pτ 2 q. Let ϕ denote the concatenation of the identity from T 1 to T pτ 2 q and the quotient map from T pτ 2 q to T 2 . By the previous observations, this map is a bijection and induces a bijection between V pT 1 q and V pT 2 q. By definition ϕ is continuous and hence shows that T 1 is isomorphic to T 2 . Proof. For ease of notation we may assume without loss of generality that T 1 " T 2 .EpT 1 q and that ι T 2 e pjq P ι T 1 e pjq for all e P EpT 1 q and j P t0, 1u. We show that τ 1 :" τ pT 1 q is isomorphic to τ 2 :" τ pT 2 q tpv, wq | v P rwsu.
Let ϕ : τ 2 Ñ τ 1 be defined as ϕpv, wq " prvs, rwsq. It is easy to see that this map is well-defined, surjective and commutes with the involution. For the injectivity consider pv 1 , w 1 q, pv 2 , w 2 q P τ 2 with v 1 P rv 2 s and w 1 P rw 2 s and let e i P EpT 2 q be such that tv i , w i u " tι T 2 e i p0q, ι T 2 e i p1qu for i P t1, 2u. Since rv 2 s and rw 2 s are both connected (as subspaces of T 2 ) but in different components of T 2´ei , we obtain that e 1 " e 2 and hence pv 1 , w 1 q " pv 2 , w 2 q.
Consider a pseudo-arc P pv, wq in T 2 between any vertices v and w. It is not hard to verify that the unique pseudo-arc in T 1 between rvs and rws has as its point set trxs P T 1 | x P P pv, wqu. This observation implies that ϕ is order-preserving and hence an isomorphism by Lemma 2.6. §5. Proof of Theorem 4.4 Now we turn to the proof of Theorem 4.4. The backwards implication is clear as pseudo-arcs are connected.
For the remainder of this section let G be a compact connected graph-like space and a and b two vertices of G.
The strategy of the proof of the forward implication is as follows. Given vertices a and b which we want to connect with a pseudo-arc, first we find a minimal set L of edges which meets every a-b-cut (that is, every cut of G that separates a and b). We then want to show that the closure of these edges in G is the desired pseudo-arc. By minimality for every edge e P L there is a signature cut, that is, an a-b-cut for which e is the only cross-edge of L. This allows us to define a linear order on L: to compare two edges e, f P L we check on which side of e's signature cut f lies. By extending this order to the points in the closure of L in G we can perform finite-intersection-arguments for suitable initial segments in order to prove connectedness.
We start off with a technical lemma that allows us to work with 'tidy' versions of our a-b-cuts. It also establishes that all topological cuts are finite if G is a compact graph-like space, which is important for the application of Zorn's Lemma. This lemma justifies the following formal definition of an a-b-cut.
A pair pA, Bq of disjoint open sets in G is an a-b-cut if: (i) a P A and b P B; (iii) for every edge e P EpGq with both end-vertices in A we havee P A; (iv) for every edge e P EpGq with both end-vertices in B we havee P B.
That is, pA, Bq is a cut separating a and b which is 'clean' in the sense of Lemma 5. This is non-empty as there is a C P C a,b which is non-empty by the connectedness of G.

Now let
Since X P L, this set is non-empty as well. We order the elements of L by inclusion. For any descending chain pM i P L | i P Iq the set M :" Ş iPI M i is a lower bound in L: for each C P C a,b every M i contains at least one edge of C, but as C is finite, so does M . Therefore Zorn's Lemma implies the existence of a minimal element L P L. We show that L is the set of edges of a pseudo-arc joining a and b.
For an edge e P L a C P C a,b is a signature cut of e if L X C " teu. In that case we also call open disjoint sets pA, Bq inducing C a signature cut of e. Such a cut exists for every e P L by the minimality of L.
Note that if pA, Bq is a signature cut of an edge e P L, then for any other f P L either For an edge e P L with end-vertices x ‰ y and a signature cut pA, Bq of e we say that e runs from x to y if x P A and y P B.
For two edges e, f P L we set e ă f if there is a signature cut pA, Bq of e withf Ď B.
Furthermore, we set e ď e for all edges e P L.
Before proceeding we need to check that neither the orientation of an edge e P L nor the definition of e ă f depends on the signature cut at hand, and that ď is a linear order on L. The general strategy in the following proofs is this: assume a counterexample to the claim exists. Consider the signature cuts of all edges involved, then for a contradiction find a suitable corner or union of corners of these cuts that is still an a-b-cut but contains no edge of L. Proof. Suppose there is an edge e P L with end-vertices x, y and signature cuts pA 1 , B 1 q and pA 2 , B 2 q, for which x P A 1 X B 2 and y P A 2 X B 1 . But then pA 1 X A 2 , B 1 Y B 2 q would induce an a-b-cut containing no edge of L: all edges of L apart from e have both their end-vertices either in B 1 Y B 2 or in A 1 X A 2 , and e has no end-vertex in A 1 X A 2 . This contradicts the definition of L. Hence x P A and y P B for all signature cuts pA, Bq of e. Now suppose there are edges e, f P L and signature cuts pA 1 , B 1 q, pA 2 , B 2 q of e such that f P B 1 X A 2 . Let pA 3 , B 3 q be a signature cut of f . Ife Ď A 3 , then the biparti- Proof. It is reflexive: this is true by definition.
Every two edges of L are comparable: suppose there are two distinct edges e, f P L with respective signature cuts pA 1 , B 1 q and pA 2 , B 2 q, for whiche Ď A 2 andf Ď A 1 . Then Then Finally we define the pseudo-arc that shall join a and b. Write L for L :" ď te | e P Lu.
As G is compact L is a compact subspace of G. Furthermore the removal of any edge e P L from L (that is, removal ofe) separates a and b in L as any signature cut of e witnesses.
To prove that L is connected we perform finite-intersection arguments on suitable initial segments of L. In order for this to be possible we first need to extend the order ď on L to an order ă on L.
Let pA, Bq be a signature cut of some e P L and x P L e. Then we write x ă e if x P A, and x ą e if x P B. For x, y P L we write x ĺ y if any of the following holds: (i) there are edges e, f P L with x Pe, y Pf and e ă f ; (ii) there is an edge e P L with x ă e ă y; (iii) there is an edge e P L with end-vertices v, w, running from v to w, such that x, y Pe and ι´1pxq ă ι´1pyq in the parametrization ι of e with ιp0q " v and ιp1q " w.
In addition we set x ĺ x for all x P L.
As for ď we prove in the following lemma that ă is well-defined in the sense that x ă e implies x P A for all signature cuts pA, Bq of e. As one readily checks ĺ is a partial order on L. If x, y P L are incomparable then x and y are both vertices that are not the end-vertex of any edge in L. To show that L is a pseudo-arc from a to b we need to show that any two vertices x, y P L are separated in L´e for some e P L. That is, we need to show that ĺ is a linear order on L. We shall achieve this with a finite intersection property argument for initial segments of L.
Let C P C a,b be some a-b-cut and LpCq :" L X C " te 1 , . . . , e n u with e 1 ă¨¨¨ă e n . For k P rn`1s the k-th segment of L with regard to C is the set S C pkq :" tx P L | e k´1 ă x ă e k u for 1 ă k ă n`1, and S C p1q :" tx P L | x ă e 1 u as well as S C pn`1q :" tx P L | x ą e n u.
As in the analogous scenario with paths and cuts in graphs one would expect the segments of L with regard to an a-b-cut pA, Bq to alternate between being contained in A or in B.
The next lemma shows that this is the case, and helps locate an edge which separates two given vertices in L.
Lemma 5.5. Let C P C a,b be induced by pA, Bq with LpCq " te 1 , . . . , e n u and e 1 ă¨¨¨ă e n .
For k P rn`1s the following statements hold.
(i) If k is odd then S C pkq Ď A; (ii) If k is even then S C pkq Ď B.
In particular, if an edge e k P LpCq has end-vertices x, y with x ĺ y, then e k runs from x to y if k is odd and from y to x if k is even.
Proof. For clarity we only consider the case where k is odd; the other case follows analogously.
First assume that k " 1. Suppose for a contradiction that there is an x P S C p1q with x P B. Let pA 1 , B 1 q be a signature cut of e 1 . Then x P B X A 1 as x ă e 1 . Due to x P L there has to be an edge f P L withf X pB X A 1 q ‰ ∅. This impliesf Ď B X A 1 and in particular e 1 ‰ f . Let pA f , B f q be a signature cut of f . Then pA X is an a-b-cut not containing any edge of L: suppose g P L is an edge with end-vertices v, w such that v P A X A 1 X A f and w P B Y B 1 Y B f . Then w P A 1 X A f implying w P B and thus g P LpCq, but also g ă e 1 , a contradiction.
If k ą 1, then suppose for a contradiction that there is an x P S C pkq with x P B.
Let pA k´1 , B k´1 q and pA k , B k q be signature cuts of e k´1 and e k respectively. Then x P B X B k´1 X A k as e k´1 ă x ă e k . Due to x P L there has to be an edge f P L with f X pB X B k´1 X A k q ‰ ∅. This impliesf Ď B X B k´1 X A k and in particular f ‰ e k´1 , e k .
Let pA f , B f q be a signature cut of f . Theǹ s an a-b-cut not containing any edge of L: suppose g P L is an edge with end-vertices v and w such that v P pB k´1 X B f q X pA Y pB X B k qq and w P A k´1 Y A f Y pB X A k q. Then w P B k´1 X B f and therefore w P B X A k , implying v P A k and thus v P A. Hence g P LpCq but e k´1 ă g ă e k , a contradiction.
Lemma 5.5 indeed implies that any two vertices of L can be separated by some e P L.
Lemma 5.6. Let v ‰ w be two vertices in L. Then there is an edge e P L which separates v and w in L.
Proof. If C is an a-b-cut with v and w on different sides, then by Lemma 5.5 v and w are in different segments, S C pk v q and S C pk w q, say. For k :" mintk v , k w u the edge e k P LpCq separates v and w in L: as x ă e ă y for any signature cut pA, Bq of e we have x P A and y P B, which gives a partition of L e into two relatively open sets.
It is thus left to show that an a-b-cut with v and w on different sides exists. Let pA, Bq be any a-b-cut and pV, W q be a v-w-cut. If v and w are on different sides of pA, Bq or if pV, W q is an a-b-cut we are done. If not, then v, w P A and a, b P V , say. But then pA X V, B Y W q is the desired cut.
From this it follows that ĺ is in fact a linear order on L. Next we prove that a P L (which, surprisingly, is not obvious) by finding a minimum of L and showing that this minimum has to be a.
Note that for any vertex c ‰ a there is an a-b-cut with c on the b-side: let pA, Bq be an a-b-cut and pA 1 , Cq be an a-c-cut. Then pA X A 1 , B Y Cq is the desired cut. Proof. We only show this for a.
If L has a minimum m P L, let a 1 be the smaller one of its end-vertices (that is, m runs from a 1 to its other end-vertex). Then a 1 is the minimum of L by Lemma 5.6.
Suppose a ‰ a 1 . Let C be an a-b-cut induced by pA, Bq with a 1 P B. Then a 1 R S C p1q, so e 1 ă a 1 implying e 1 ă m a contradiction to the minimality of m.
If L does not have a minimum then for e P L set X e :" ď tf | f P L, f ă eu.
Then X e Ď L for all e P L. Since G is compact, L has the finite intersection property. Therefore X :" For any edge e P L no inner point x Pe of e is in X, as x R X e . Thus X contains a vertex a 1 .
If there were another vertex a 2 P X, then a 1 and a 2 could be separated by an edge e P L by Lemma 5.6 and one of them would not be in X e . So X " ta 1 u. Suppose a ‰ a 1 . Let C be an a-b-cut induced by pA, Bq with a 1 P B and let LpCq " te 1 , . . . , e n u with e 1 ă¨¨¨ă e n .
Then a 1 R S C p1q as a 1 P B, so e 1 ă a 1 . But this means a 1 R X e 1 , a contradiction.
The final property needed of L to be a pseudo-arc joining a and b is that it is connected.
The proof of this is similar to the proof of Lemma 5.7. We aim to find a minimum of Y " S with regard to ĺ.
If S has a minimum m P S with regard to ď then let y be the smaller one of its end-vertices. Then y P Y and y ĺ z for all z P S.
If S does not have a minimum then for e P S set R e :" ď tf | f P S, f ă eu.
Every R e is a non-empty closed subset of L. By the finite intersection property R :" Ş ePS R e is non-empty. For any edge e P S no inner point x Pe of e is in R, as y R R e . Thus R contains a vertex y. If there were another vertex y 1 P R, then y and y 1 could be separated by an edge e P L by Lemma 5.6, with y ă e ă y 1 , say. This edge e cannot be in S as in that case y would not be in R e . Thuse Ď X. Let pA, Bq be a signature cut of e. As e ă f for all f P S due to e ă y 1 ă f we have y P A and ď tf | f P Su Ď B.
But then A X L witnesses that y R S, a contradiction.
Therefore R " tyu and y is the minimum of S. Now set X 1 :" tx P X | x ă e for all e P Su and let U :" te P L |e Ď X 1 u. By a similar argument as above X 1 has a maximum x.
Let y be the minimum of Y " S and e P L an edge separating x and y. If y ă e ă x then either e P S and x R X 1 or e P U and y R Y . So x ă e ă y, which implies e P U . But this contradicts the fact that x is the maximum of X 1 .
We have succeeded in proving that L is a pseudo-arc containing a and b. This concludes our proof of Theorem 4.4. l