PROJECTIVE FRAÏSSÉ LIMITS AND GENERALIZED WAŻEWSKI DENDRITES

. We continue the study of projective Fraïssé limits of trees initiated by Charatonik and Roe and we construct many generalized Ważewski dendrites as the topological realization of a projective Fraïssé limit of families of ﬁnite trees with (weakly) coherent epimorphisms. Moreover we use the categorical approach to Fraïssé limits developed by Kubiś to construct all generalized Ważewski dendrites as topological realizations of Fraïssé limits of suitable categories of ﬁnite structures. As an application we recover a homogeneity result for countable dense sets of endpoints in generalized Ważewski dendrites.


Introduction
Projective Fraïssé limits were introduced by Irwin and Solecki in [13] by dualizing the classical construction of Fraïssé limits from model theory, as a tool to study the pseudo-arc.Panagiotopolous and Solecki generalised their construction in [22] by allowing strict subcollections of epimorphism, instead of considering all of them.They used this approach to study the Menger curve by approximating it with finite graphs with monotone epimorphisms.Projective Fraïssé limits have since been used to study various compact spaces, for example the Lelek fan by Bartošová and Kwiatkowska [3], the Poulsen simplex by Kubiś and Kwiatkowska [17], P -adic pseudo-solenoids (for a set of primes P ) by Bartoš and Kubiś [2], the universal Knaster continuum by Iyer [14] (which has independently been constructed from a projective Fraïssé limit in Wickman's PhD thesis [23]) and the generic object for a new class of compact metric spaces named fences by Basso and Camerlo [4].Recently Charatonik, Kwiatkowska, Roe and Yang studied projective Fraïssé limits of trees with monotone and confluent epimorphisms, which are adaptations to topological graphs of corresponding well established notions from continuum theory, see [6].
Dendrites are one-dimensional continua which have been widely studied in topological dynamics, [9], [10], [1], [20], [11].By a result of Charatonik and Dilks [5], for any P ⊆ {3, 4, . . ., ω}, there exists a unique dendrite W P such that: (1) every ramification point of W P has order in P ; (2) for every p ∈ P the set of ramification points of W P of order p is arcwise dense in W P .
The spaces W P are known as the (generalized) Ważewski dendrites, they are universal for appropriate classes of dendrites and enjoy good homogeneity properties.Because of these regularity properties they are of particular interest among dendrites.Their homeomorphism groups have been studied by Duchesne [8], while the universal minimal flows of those groups were computed by Kwiatkowska [19].Among the spaces constructed in [6] there is the Ważewski dendrite W 3 (denoted by D 3 in their paper), obtained from the projective Fraïssé limit of the family of finite trees with ramification points of order 3 and all monotone epimorphisms; we build on their work to realize many generalized Ważewski dendrites W P for P ⊆ {3, 4, 5, . . ., ω} as the topological realizations of projective Fraïssé limits of appropriately chosen families of finite trees and epimorphisms between them.
In particular, we obtain the following theorem.See Definition 4.14, the remarks following it, and Theorems 4.19 and 4.20 for details.
Theorem 1.1.Let P ⊆ {3, 4, . . ., ω} be coinfinite.Then the Ważewski dendrite W P can be constructed as the topological realization of the projective Fraïssé limit of finite trees with monotone maps with additional properties.
By moving to the more general setting of Fraïssé categories developed in [16] we remove the coinfiniteness assumption from the previous result and obtain the following general statement, see Theorems 5.10 and 5.23 for details.
Theorem 1.2.Let P ⊆ {3, 4, . . ., ω} and let E ⊆ W P be a countable dense set of endpoints.Then the pair (W P , E) can be constructed as the topological realization of the limit of a Fraïssé category ‡F P of pairs (T, F ) with projection-embedding pairs of maps, where T is a finite tree and F is a set of endpoints in T .
Moreover we show in Theorem 5.22 that any sequence in the category ‡F P whose limit has as topological realization W P , must be a Fraïssé sequence for ‡F P .
A Fraïssé theoretic approach to the study of Ważewski dendrites has already been employed successfully by Kwiatkowska in [19], in which a Fraïssé structure A P with Aut(A P ) ≃ Homeo(W P ) is constructed.While this approach is well suited to study properties of Homeo(W P ), it cannot be used to study the endpoints of W P , since the injective Fraïssé limit construction produces a countable structure corresponding to the ramification points of W P .In contrast, we use the uniqueness of Fraïssé sequences in the last section to recover a homogeneity result for the endpoints of W P from [5], see Theorem 5.11.
The paper is structured as follows.In Sections 2 and 3 we recall some background notions concerning projective Fraïssé limits and topological graphs.In Section 4 we introduce a new class of maps between finite trees and use it to prove Theorem 1.1.Finally in Section 5 we describe projectionembedding Fraïssé categories of finite trees that can be used to prove Theorem 1.2.

Projective Fraïssé Limits
In this section we recall the necessary background on projective Fraïssé limits.Most definitions and results are from either [13] or [22].Definition 2.1.Let L be a first order language consisting of a set {R i } i∈I of relation symbols, with R i of arity n i ∈ N, and a set {f j } j∈J of function symbols, with f j of arity m j .A topological L-structure is a zero-dimensional, compact, metrizable topological space A together with closed subsets R A i ⊆ A n i for every i ∈ I and continuous functions f A j : A m j → A for every j ∈ J.
Definition 2.2.Given two topological L-structures A and B we say that g : B → A is an epimorphism if g is a continuous surjection satisfying (1) For all j ∈ J, f A j (g(b 1 ), . . ., g(b m j )) = g(f B j (b 1 , . . ., b m j )) (2) For all i ∈ I, we have R A i (a 1 , . . ., a n i ) if and only if there are b 1 ∈ g −1 (a 1 ), . . ., b n i ∈ g −1 (a n i ) such that R B i (b 1 , . . ., b n i ) holds.
We can now give the definition of a projective Fraïssé family and its projective Fraïssé limit.We follow the construction from [22].The main difference compared to the original construction from [13] is allowing a restricted class of epimorphisms between the structures under consideration.Definition 2.3.Let F be a class of finite topological L-structures with a fixed class of epimorphisms between them.We say that F is a projective Fraïssé family if it satisfies the following conditions: (1) Up to isomorphism there are countably many L-structures in F .(2) The epimorphisms in F are closed under composition and the identity epimorphisms are in F .(3) For all A, B ∈ F there are C ∈ F and epimorphisms f : C → A and g : C → B in F .In analogy with the classical Fraïssé limits this property is known as the joint projection property.(4) For all A, B, C ∈ F and epimorphisms f : B → A and g : C → A in F , there exist D ∈ F and epimorphisms h : D → B and k : D → C in F such that f • h = g • k.This property is known as the projective amalgamation property.
Following [22] we introduce F ω as the class of structures that can be approximated in F , indeed the projective Fraïssé limit will be an element of F ω satisfying appropriate universality and homogeneity properties.Definition 2.4.Let F be a class of finite topological L-structures.We define F ω to be the class of all topological L-structures in F that are realized as the inverse limits of an ω-indexed inverse system of topological L-structures with bonding epimorphisms in F .More explicitly F ∈ F ω iff there is an inverse system of topological L-structures F i , f n m : F n → F m , i, n, m < ω with F i , f n m ∈ F , such that F = lim ← − F i (note in particular that F ⊆ F ω by picking constant sequences).
Note that, given an inverse system F i , f n m : F n → F m , i, n, m < ω in F as above, F = lim ← − F i exists.Its underlying space is the inverse limit of the F i in the category of topological spaces, while for f n : F → F n , the continuous maps obtained by construction of the limit of topological spaces, we have and Theorem 2.5 ([22,Theorem 3.1]).Let F be a projective Fraïssé family of topological L-structures.Then there exists a unique topological L-structure F ∈ F ω satisfying the following properties: (1) For every A ∈ F there is an epimorphism f : F → A in F ω .
(2) For all A, B ∈ F ω and all epimorphisms f : F → A and g : B → A in F ω , there exist an epimorphism h : Moreover, as pointed out in [13, Lemma 2.2] in terms of covers, we also have that F satisfies (3) For every ε > 0 there is A ∈ F and an epimorphism f : Proof.We give a sketch of the argument, the details can be found in [13] and [22].The idea is to obtain F as the inverse limit of a sequence elements and epimorphisms of F .We will call this sequence a Fraïssé sequence for F .Since there are only countably many finite structures in F up to isomorphism, we can build two countable lists Inductively assume that F n together with the epimorphisms f n m : F n → F m for all m < n have been defined.By the joint projection property we can find H ∈ F with epimorphisms g : H → A n+1 and f : H → F n .Since H is finite there are only finitely many epimorphism types s i : H → C n+1 , i = 1, . . ., k.Using the projective amalgamation property, we can find H ′ 1 and epimorphisms h 1 : It is easy to verify that F = lim ← − F i satisfies the first two properties of a projective Fraïssé limit of F .The third one follows from the definition of the product metric on F i .
From the proof of Theorem 2.5 we extract the following important notion.Definition 2.6.Let F be a projective Fraïssé family of topological Lstructures.A sequence • for every A ∈ F there exists n ∈ N such that there is an epimorphism f : such that there is an epimorphism g :

Topological Graphs and Monotone Epimorphisms
In this section we reproduce for convenience definitions and results about topological graphs that will be used in Section 4 and Section 5.They are mostly taken from [6].
Theorems 3.7, 3.14, 3.15, 3.16, 3.17 first appeared in the preprint of Charatonik-Roe [7], but since that preprint is unpublished, with the permission of the authors, we include the proofs for completeness.Those results will appear in [6].

Topological Graphs and epimorphisms
A finite graph T is a tree if there are no pairwise distinct vertices a 1 , . . ., a n ∈ T , n ≥ 3, such that a i , a i+1 ∈ E(T ) for i = 1, 2, . . ., n − 1, and a n , a 1 ∈ E(T ).We will use interchangeably a ∈ V (A) and a ∈ A and we call the edges a, b ∈ E(A) with a = b nontrivial.A topological graph is a graph G equipped with a compact, Hausdorff, zero-dimensional, metrizable topology on V (G) so that E(G) is a closed subspace of V (G) 2 .Every topological graph is a topological L-structure for the language L = {R} consisting of a single binary relation in an obvious way.We will keep using the notation a, b ∈ E(G) rather than aRb even though we will often think about topological graphs as topological L-structures.An epimorphism f : A → B of topological graphs is an epimorphism in the sense of Definition 2.2, so in particular it maps edges to edges.
Let us point out that in this context requiring topological graphs to be zero-dimensional is a natural property, since we are interested in inverse limits of finite discrete structures.We warn the reader that in the literature there are different definitions of topological graphs, where edges are embedded as arcs into the topological structure.
If A is a topological graph and B ⊆ A, A \ B denotes the topological graph with It is called a prespace if the edge relation aRb ⇐⇒ a, b ∈ E(G) is an equivalence relation, equivalently a prespace is a topological graph with a transitive set of edges.In that case we call the quotient topological space  Then G is a prespace, in particular |G| ∼ = [0, 1].The notion of a monotone map from continuum theory, where a continuous map f : X → Y between continua is called monotone if f −1 (y) is connected for every y ∈ Y , was adapted to topological graphs by Panagiotopoulos and Solecki in [22] (where those maps were called connected epimorphism instead) and studied in detail by Charatonik, Kwiatkowska, Roe, and Yang in [6].The epimorphisms we will consider in the following sections will always be monotone so we will use various results from [6], whose statements and proofs we include here for completeness.Definition 3.5.An epimorphism f : G → H between topological graphs is called monotone if f −1 (h) is connected in G for every h ∈ H.
By Lemma 1.1 of [22] we can equivalently require that f −1 (C) is connected in G whenever C ⊆ H is closed connected in H. From this equivalent definition it is clear that if f : G → H is monotone and H is connected then G is too.The converse of this statement actually holds for arbitrary epimorphisms, an observation repeatedly used in both [22] and [6]: Remark 3.6.Let f : G → H be an epimorphism between topological graphs.If G is connected then H is connected as well.
Theorem 3.7 ([6, Theorem 2.17]).Let G be a projective Fraïssé family of graphs such that for every G ∈ G and a, b, c ∈ G pairwise distinct with a, b , b, c ∈ E(G), there is a graph H ∈ G and an epimorphism f H G : H → G so that whenever p, q, r ∈ H are such that f H G (p) = a, f H G (q) = b and f H G (r) = c, we have p, q ∈ E(H) or q, r ∈ E(H).Then, if G denotes the projective Fraïssé limit of G, for every a ∈ G there is at most one b ∈ G\{a} with a, b ∈ E(G).In particular, the edge relation of G is an equivalence relation, that is, G is a prespace.
Proof.Suppose on the contrary that there are three distinct vertices a, b, c ∈ G such that a, b , b, c ∈ E(G).Let a graph G ∈ G and an epimorphism This contradicts the fact that f H maps edges to edges.Definition 3.8.Let T be a projective Fraïssé family of finite graphs.We say that T allows splitting edges if for all G ∈ T and all distinct a, b ∈ G with a, b ∈ E(G) the graph H defined by V (H) = V (G) ⊔ { * } and and the two morphisms H → G that map * to either a or b and are the identity otherwise, are in T .
Allowing to split edges is a basic closure property that will be satisfied by all the projective Fraïssé families considered in what follows.Despite its simplicity it has important consequences, such as the following lemma, and it will be used for some arguments in the next sections.Lemma 3.9.Suppose that T is a projective Fraïssé family of finite graphs that allows splitting edges.If G is the projective Fraïssé limit of T , then G is a prespace.
Proof.We will show that T satisfies the hypothesis of Theorem 3.7, from which the conclusion follows immediately.Fix G ∈ T and pairwise distinct a, b, c ∈ G with a, b , b, c ∈ E(G) and let H be the graph obtained from G by splitting the edge a, b twice.More formally we have and f H G is the identity otherwise.We have that H and f H G are in T since T allows splitting edges, but if p, q, r ∈ H are such that f H G (p) = a, f H G (q) = b and f H G (r) = c, then either p, q ∈ E(H), or q, r ∈ E(H).

3.2.
Dendrites.We begin this section by recalling some facts concerning dendrites from general topology.A compact connected metrizable space is called a dendrite if it is locally connected and contains no simple closed curve [21,Definition 10.1].Equivalently dendrites are locally connected dendroids [12,Page 218].For more characterizations of dendrites see [21,Section 10.1].Therefore a dendrite X is uniquely arcwise connected, that is between any x = y ∈ X, there is a unique arc.In a dendrite X there are three types of points x ∈ X, based on how many connected components X \ {x} has: • If X \ {x} is connected, then x is called an endpoint of X.
• If X \ {x} has two connected components, then x is called a regular point of X. • If X \{x} has more than two connected components, then x is called a ramification point of X.
In all of those cases the (potentially infinite) number of connected components of X \ {x} is called the order or degree of x, denoted by ord(x), and it coincides with the Menger-Urysohn order of x in X, that is with the maximum number of arcs in X meeting only in x [18, §46, I].For P ⊆ {3, 4, 5, . . ., ω} the generalized Ważewski dendrite W P is a dendrite satisfying the following two properties: • If x ∈ W P is a ramification point, ord(x) ∈ P .
• If p ∈ P , the set of ramification points in W P of order p is arcwise dense.This means that for all x, y ∈ W P distinct, there is a ramification point of order p in the unique arc joining x and y.
For a fixed P those two properties characterize a unique dendrite up to homeomorphism [5, Theorem 6.2], so we can actually talk about the generalized Ważewski dendrite W P .These notions were adapted to topological graphs in [6], from which the remaining definitions and results of this section are taken.Definition 3.10.A topological graph A is called arcwise connected if for all a, b ∈ A, there is a subgraph of A containing a and b which is an arc.A topological graph A is called locally connected if every a ∈ A has an open neighbourhood which is connected.A topological graph G is hereditarily unicoherent if for every two non-empty closed connected topological graphs, P and Q, with The topological graph G is unicoherent if the above holds for all P and Q such that Definition 3.11.A topological graph G is called a graph-dendroid if it is hereditarily unicoherent and arcwise connected.A graph-dendroid G is called a graph-dendrite if it is locally connected.Definition 3.12.Let G be a topological graph.A vertex x ∈ G is called an endpoint of G if whenever H is a topological graph which is an arc and f : H → G is an embedding with x ∈ f (H), then x is in the image of an endpoint of H.Note that, when G is an arc, this notion agrees with the one in the Definition 3.3.Definition 3.13.Let X be a graph-dendrite.We say that x ∈ X is a ramification point if X \ {x} has at least 3 components, in which case we call the (possibly infinite) number of components of X \ {x} the order of x in X.
Theorem 3.14 ([6, Lemma 3.7]).If f : G → H is a monotone epimorphism between topological graphs and G is an arc, then H is an arc and the images of endpoints of G are endpoints of H.
Proof.Denote the end vertices of G by a and b.We need to show that every vertex in Our goal is to construct a large class of dendrites from projective Fraïssé limits of finite graphs.The following results from [6] will be crucial.Lemma 3.15 is only stated in [6], below we supplement a proof.Lemma 3.15 ([6,Lemma 3.8]).Let T be the inverse limit of finite arcs with monotone epimorphisms.Then T is an arc.
Proof.Let I n | n < ω with monotone epimorphisms f m n : I m → I n be such that T = lim ← − I n , and let f n : T → I n be the induced projections.Note that, since monotone epimorphisms map endpoints to endpoints, there are two points a = (a n ) and b = (b n ) in T such that a n , b n are endpoints of I n for every n.In particular for any x ∈ T \ {a, b} there is N < ω such that We want to show that every x ∈ T \ {a, b} disconnects T. Let N be as above for x and let, for k ≥ N, U k a and U k b be the connected components of (and analogously for b) so that Theorem 3.16 ([6, Theorem 2.18]).Let G be the inverse limit of trees with monotone epimorphisms.Then G is hereditarily unicoherent.
The proof below is an adaptation of the proof in Nadler [21,Theorem 10.36] of the fact that the inverse limit of dendrites is hereditarily unicoherent.It is different from the proof given in [6].
C n = ∅ for some n there is nothing to prove, so we can assume that C n = ∅ for all n.Since every G n is a tree, hence hereditarily unicoherent, C n is connected and nonempty for every n.It is now not hard to check that This implies that C is connected, since it is the inverse limit of closed connected sets.
Theorem 3.17 ([6, Corollary 3.11+Observation 2.16]).Let T be a projective Fraïssé family of trees with monotone epimorphisms, let G be its projective Fraïssé limit, then G is a graph-dendrite.Moreover if G has a transitive set of edges, then |G| is a dendrite.
Proof.It is already proved in [22, Proposition 2.1] that G is connected and locally connected, and it is proved in [22, Theorem 2.1] that |G| is a Peano continuum (i.e. it is a locally connected compact connected space).
Peano continua are arcwise connected [21,Theorem 8.23].It follows from Lemma 3.15 that G is arcwise connected (see also [6,Proposition 3.10]), and it follows from Theorem 3.16 that G is hereditarily unicoherent.The argument by contradiction can be used to show that |G| is hereditarily unicoherent whenever G is, which concludes the proof.
Note that combining Lemma 3.9 and Theorem 3.17 we have that all projective Fraïssé families of trees with monotone maps that allow splitting edges have as limit a prespace whose topological realization is a dendrite.In particular this will be true for all projective Fraïssé families considered in the following sections.The result below, which is contained in [22, Proposition 2.1], will also be important.Lemma 3.18.Let T be a projective Fraïssé family of graphs with monotone epimorphisms.Let T i | i < ω with maps f i+1 i : T i+1 → T i be a Fraïssé sequence for T and let T be its projective Fraïssé limit.Then for G ∈ T all epimorphisms T → G in T ω are monotone.
Proof.Since epimorphisms f i+1 i are monotone, it is not hard to see that the projection epimorphisms f i : T → T i are monotone.Any epimorphism T → G is a composition of a projection epimorphism f i , for some i, and a monotone epimorphism from T i to G.
In this section, we introduce a new class of maps between trees, which we call (weakly) coherent and study the relationship between weakly coherent points in a prespace and ramification points of its topological realization.Once this is done we will show how to construct many generalized Ważewski dendrites as the topological realization of a projective Fraïssé limit.
In this case we call b the witness for the weak coherence of f at a. We say that f is weakly coherent if every a ∈ A with ord(a) ≥ 3 is a point of weak coherence for f .This notion can be strengthened by requiring ord(b) = ord(a) and p to be a bijection.In that case we say that a is a point of coherence of f and that b is a witness for the coherence of f at a. We say that f is coherent if every a ∈ A with ord(a) ≥ 3 is a point of coherence for f .Remark 4.2.Definitions of a weakly coherent and of a coherent epimorphism can be extended (in the obvious way) to epimorphisms between inverse limits of trees.The map f defined by f (a 1 ) = f (a 2 ) = a and mapping every other node to the node with the same name is a monotone epimorphism which is not weakly coherent at a.
) and let Q be the shortest arc from b ′ 3 to a point in P , let p i = P ∩ Q and note that, by another application of Remark 3.6, similar to the one above, It is now easy to check that p i witnesses the coherence of f at b.  Definition 4.9.Given an inverse system A i , i < ω of graphs and monotone epimorphisms f i j : A i → A j we say that a = (a i ) i ∈ lim ← − A i ⊆ A i is a point of weak coherence or a weakly coherent point if there is a k ∈ ω such that for all l > k, a l is a point of weak coherence of f l+1 l and a l+1 witnesses this.We say that a is a point of coherence or a coherent point if there is a k ∈ ω such that for all l > k, a l is a point of coherence of f l+1 l , and a l+1 witnesses this.Lemma 4.10.Let T be a projective Fraïssé family of finite trees with monotone epimorphisms that allows splitting edges, let G be its projective Fraïssé limit and let π : G → |G| be its topological realization, which is a dendrite by Theorem 3.17.Moreover let p ∈ |G| be a ramification point.Then there is a unique p ′ ∈ G with π(p ′ ) = p.
Proof.Suppose for a contradiction that |π −1 (p)| > 1 and, using that equivalence classes in G have at most two elements by Theorem 3.7, write π −1 (p) = . This is possible because epimorphisms in T ω are monotone by Lemma 3.18.We claim that The idea now is that we can always move to a bigger graph H obtained from G by splitting an edge in order to force f where r is a vertex not in G and Note that H, f H G ∈ T since T allows splitting edges.By definition of projective Fraïssé limit there must be an epimorphism The last identity combined with the fact that f H maps edges to edges forces f H (p 1 ) = a and f H (p 2 ) = r.Now we can use the same argument as above to show that This is a contradiction, since by construction those two sets also meet in b.
Theorem 4.11.Let T be a projective Fraïssé family of trees with monotone epimorphisms that allows splitting edges, let G i | i < ω with epimorphisms f m n : G m → G n be a Fraïssé sequence for T , let G = lim ← − G i be the projective Fraïssé limit of T and let π : G → |G| be its topological realization, which is a dendrite by Theorem 3.17.Then π maps points of weak coherence of G ⊆ G i to ramification points of |G|.
Proof.Suppose that p = (p i ) is a point of weak coherence of G, we want to show that π(p) is a ramification point of |G|.Let f k : G → G k be the projection epimorphisms obtained by construction of the inverse limit, let We will show that the equivalence class of p in G is a singleton, from which we can conclude that π(A i ) are the connected components of |G|\{π(p)}, so that π(p) is a ramification point of |G| of order equal to the least n such that k(n + 1) = ω or to ω if there is no such n.Suppose for a contradiction that there exist q = (q i ) ∈ G with p, q ∈ E(G), so in particular p i , q i ∈ E(G i ) for all i.Let k be big enough so that p k = q k and let A j k be the component of G k \ {p k } containing q k .Note that by weak coherence for all m ≥ k, q m ∈ A j m .Consider the tree H obtained from G k by splitting the p k , q k edge into two edges p k , p ′ k and p ′ k , q k , with the epimorphism  Remark 4.12.Note in particular that if x = (x i ) is a point of weak coherence in G i then either ord(x i ) stabilizes after some index j ∈ N to some finite value n or it grows unboundedly.In the former case π(x) also has order n as a ramification point of |G|, while in the latter π(x) is a ramification point of |G| of infinite order.
The converse to Theorem 4.11 is false in general, but it is true when monotone epimorphisms are replaced with weakly coherent ones, as shown in the following theorem.Theorem 4.13.Let T be a projective Fraïssé family of trees with weakly coherent epimorphisms, let G i | i < ω with epimorphisms f m n : G m → G n be a Fraïssé sequence for T and let π : G → |G| be its topological realization, which is a dendrite by theorem 3.17.Then π −1 maps ramification points of |G| to points of weak coherence of G ⊆ G i .Proof.As above let f k : G → G k be the projection epimorphisms obtained by construction of the limit and let x ′ ∈ |G| be a ramification point.By Lemma 4.10 there is a unique x = (x i ) ∈ π −1 (x ′ ) and we want to show that it is a point of weak coherence in ) is connected and doesn't meet f −1 k+1 (x k+1 ).Since x k+1 = a and G k+1 is a tree, there is a connected component A n k+1 of G \ {x k+1 }, that contains all connected components of G \ {a} except at most one, so it also contains (f k+1 k ) −1 (A i k ) for at least two distinct i. Assume without loss of generality that , and let j be the unique index such that , which cannot all hold at the same time regardless of the value of j, a contradiction.

Generalized Ważewski Dendrites from Projective Fraïssé Families.
We now introduce some families of finite trees.The remainder of this section will be dedicated to showing that those are projective Fraïssé families and that the topological realizations of their projective Fraïssé limits are generalized Ważewski dendrites.Definition 4.14.Let P ⊆ {3, 4, 5, . . ., ω}.We consider two cases.If ω ∈ P we consider the family F P whose elements are finite trees with no vertices of order 2. Given A, B ∈ F P , an epimorphism of graphs f : On the other hand, if ω ∈ P , we consider the family G P whose elements are finite trees all of whose vertices are either endpoints or have order in P , with coherent monotone epimorphisms.
Remark 4.15.The families F P and G P don't allow splitting edges as defined earlier, since the trees in those families don't have any vertices of degree two.However if T is a tree in F P or G P and a, b ∈ T with a, b ∈ E(T ) it is still possible to split the edge a, b by removing it and then adding a new point x connected to a and b.We have to also add enough new neighbours x i to x until ord(x) ∈ P .This is enough for all the arguments in the previous sections that used the edge splitting property.
We will now verify that F P and G P are projective Fraïssé families and later we will identify the topological realizations of their projective Fraïssé limits.It seems natural to guess that if F P and G P are the projective Fraïssé limits of F P and G P , then |F P | ∼ = W P and |G P | ∼ = W P , but unfortunately this is not always the case.It is true for all families of the form G P , but for families of the form F P it only holds when {3, 4, 5, . . ., ω} \ P is infinite, as we will see later.Clearly both F P and G P contain countably many structures up to isomorphism, contain the identity morphisms and their morphisms are closed under composition by Remark 4.7.We begin by verifying the joint projection property for F P and G P , with a construction that works for both families.
Lemma 4.16.Let A, B ∈ F P (respectively A, B ∈ G P ).Then there exists Proof.Let a ∈ A, b ∈ B be two endpoints.Consider the tree C ′ obtained by taking the disjoint union of A and B and identifying a with b.Let C be the tree obtained from C ′ by adding new vertices x i connected only to x until ord(x) ∈ P , where x is the vertex a and b have been identified in.Define f : C → A to be the identity on V (A), while any other vertex is mapped to x. Analogously define G : C → B to be the identity on V (B), while any other vertex is mapped to x.It is immediate to check that C witnesses the joint projection property for A, B.
It remains to verify that the projective amalgamation property is satisfied.We do so for families of the form F P and note that the procedure described for those to produce an amalgam D from a diagram C → A ← B of structures in F P can also be applied to diagrams of the same shape in G P , and the amalgam D produced in that case is itself an element of G P .
Lemma 4.17.The family F P satisfies the projective amalgamation property, so it is a projective Fraïssé family.Explicitly for all A, B, C ∈ F P with epimorphisms f : B → A and g : C → A in F P , there exist D ∈ F P with epimorphisms h 1 : D → B and h 2 : Proof.We proceed by induction on |E(B)| + |E(C)|.If there are no nontrivial edges in B and C then A must also be a singleton and we can take D to be a singleton as well.For the inductive step suppose now that B and C are not both singletons and that the lemma is proved for all diagrams in we can simply take D = A and h 1 = h 2 = Id A , so we can assume that at least one of f and g is nontrivial.Suppose without loss of generality that f is nontrivial.Find a ∈ A such that |f −1 (a)| > 1.
There are two possibilities now, a could be a ramification point or and endpoint of A, and we deal with those two cases separately.
Case 1: a is a ramification point.Let b ∈ B be the witness for the (weak) coherence of f at a and let B 1 be a component of B \ {b} with . We need to distinguish two cases, based on whether B 2 = B 1 or not.
Case 1a: B 2 = B 1 We start with the harder case, that is B 2 = B 1 and explain how to deal with the easier case later.Let b 1 be the only element of There is a natural map π : B → B ′ given by π(x) = b if x ∈ B 2 and the identity otherwise.Note that B ′ ∈ F P since both b and b 3 have the same order as π(b) and π(b 3 ), and π is a coherent epimorphism, so in particular it is in F P .Let f ′ : B ′ → A be the unique map with f ′ • π = f and note that f ′ is still in F P by construction.We reach the following picture, where the important vertices and edges are drawn, while the dashed sections represent parts of the trees whose precise structure is not important.Since |E(B ′ )| < |E(B)| we can amalgamate f ′ and g over A by inductive hypothesis, so we can find D ′ ∈ F P and epimorphisms h ′ 1 : 1 and h ′ 2 , the intuitive idea is that we just need to paste back B 2 in D ′ in the right spot, in particular B 2 should be inserted on the edge of D ′ that corresponds to the edge π(b), π(b 3 ) in B ′ (analogously to how B is obtained from B ′ by inserting B 2 on the π(b), π(b 3 ) edge).Let a 1 be the unique vertex of A with a, a 1 ∈ E(A) and f −1 (a 1 ) ⊆ B 1 and let A 1 be the connected component of A\ {a} containing a 1 .Let c ∈ C witness the (weak) coherence note that B 2 ∈ F P and we didn't change the degrees of d 1 and d 2 , so D ∈ F P as well.Now consider the projection π and the identity otherwise, which is also an epimorphism in F P .We are now in a situation that looks like Figure 2.
where we identify points of B \ B 2 with their image in B ′ through π.Note that h 1 is an epimorphism in F P , indeed it is (weakly) coherent at points of B 2 because they are witnesses for their own (weak) coherence and it is (weakly) coherent elsewhere because h ′ 1 is.Moreover we have h The construction of D ′ from D so that the following diagram commutes: It only remains to check that (f • h 1 )(x) = (g • h 2 )(x) for all x ∈ D, for which there are two cases: and h 2 (x) agree with h ′ 1 (x) and h ′ 2 (x) respectively, so we immediately obtain f (h 1 (x)) = g(h 2 (x)) by construction.

This concludes the argument when
Case 1b: B 2 = B 1 .Let b ∈ B be the witness for the (weak) coherence of f at a and let c ∈ C be the witness for the (weak) coherence of g at a. Suppose now that for every component B of B \ {b} that meets f −1 (a), we have B ⊆ f −1 (a), otherwise modulo choosing a different component B 1 of B \ {b} we would be in the previous case.Similarly we can assume that for every component Ĉ of C \ {c} that meets g −1 (a), we have Ĉ ⊆ g −1 (a), since otherwise we would be in the previous case modulo renaming B and C. Let a A 1 , . . ., a A n enumerate the connected components of A \ {a}.By the weak coherence of f we can find a B 1 , . . ., a B n components of B \ {b} such that f −1 (a A i ) ⊆ a B i for all i.We define a C 1 , . . ., a C n analogously.Now let

The idea now is to amalgamate the diagram (1)
a B i ∪ {b} → {a} ∪ {a A i } ← a C i ∪ {c} for every i, then to obtain D 1 from B 1 and C 1 by the joint projection property, and to glue everything back together at the end.By inductive hypothesis we can amalgamate diagrams of the form (1) (note that the trees involved are still members of F P and so are the maps) obtaining an amalgam a D i with maps h i 1 : a D i → a B i ∪ {b} and h i 2 : a D i → a C i ∪ {c}.Note that, since b and c are the only preimages of a in a B i ∪ {b} and a C i ∪ {c} respectively, there must be an endpoint We now need to deal with B 1 and C 1 .Enumerate as c 1 , . . ., c r the components of C 1 \ {c} and as b 1 , . . ., b s those of B 1 \ {b}.Assume without loss of generality s ≤ r.For i ≤ r, let c i be the only vertex of c i such that c i , c ∈ E(C) and let b i , for i ≤ s, be defined analogously.Consider now the tree D 1 built from i≤r c i ⊔ i≤s b i by adding r + 1 new vertices z, z 1 , . . ., z r and edges z i , c i for all i ≤ r, z i , b i for all i ≤ s and z, z i for all i ≤ r.
Finally build a tree D by identifying z ∈ D 1 and x i ∈ a D i for every i, and calling d the image of all those points in D, add neighbours to z i , b i and c i until they have order in P .Define a map h 1 : D → B by It is easy to check that D together with the maps h 1 , h 2 is an amalgam of B and C over A. This concludes the argument for a ramification point a ∈ A.
Case 2: a is an endpoint.Suppose now that a ∈ A is an endpoint.The construction is essentially an easier version of the one we just carried out for the case in which B 1 = B 2 .Since B \ f −1 (a) is connected we can let b ∈ f −1 (a) be the unique vertex for which there is a b ′ ∈ B \ f −1 (a) with b, b ′ ∈ E(B) and let B 2 be f −1 (a).As in the previous case consider π : B → B ′ collapsing B 2 to b and let f ′ : B ′ → A be the unique map with f ′ • π = f .Moreover let D ′ be an amalgam of f ′ and g over A with maps h Once again we build D from D ′ by glueing B 2 to D ′ in the right spot.Let α ∈ (h ′ 1 ) −1 (b) be an endpoint and take the disjoint union of B 2 and D ′ , identifying α ∈ D ′ and b ∈ B 2 .Let d be the image of α and b through this identification.
As in the previous case, we obtain a map π ′ : D → D ′ by mapping the whole of B 2 to d, and a map h 1 : D → B defined by we see that D with the maps h 2 : D → C and h 1 : D → B defined above gives an amalgam of f and g over A, as desired.
As mentioned before Lemma 4.17, the construction in its proof also shows the following: Lemma 4.18.The family G P satisfies the projective amalgamation property, so it is a projective Fraïssé family.Now that we know that F P and G P are projective Fraïssé families let F P and G P denote their projective Fraïssé limits.We know from Lemma 3.9 and Theorem 3.17 that |F P | and |G P | are dendrites, and in the rest of this section we will identify which dendrites they are exactly.Let's start with the easiest case, namely ω ∈ P , so that we're looking at the family G P with projective Fraïssé limit |G P |.Proof.For the remainder of this proof we fix a Fraïssé sequence G i for G P , with epimorphisms f i j : G i → G j .We consider G P as a subspace of G i , and call f i : G P → G i the canonical projections.We know that |G P | is a dendrite, so we need to check that all ramification points of |G P | have order in P , and that for all p ∈ P the set {x ∈ |G P | | ord(x) = p} is arcwise dense in |G P |.The first part follows quickly from previous results: if x ∈ |G P | is a ramification point, then π −1 (x) = (x i ) is a coherent point in G P by Theorem 4.13, where π : G P → |G P | is the topological realization.Note that ord(x i ) ∈ P for large enough i, so by Remark 4.12 we have that, for k big enough, ord(x) = ord(x k ) ∈ P .It remains to show that for every p ∈ P the set of ramification points in |G P | is arcwise dense in |G P |.Fix p ∈ P and X ⊆ |G| an arc.Let a ′ , b ′ ∈ X be two distinct points and fix a ∈ π −1 (a ′ ), b ∈ π −1 (b ′ ).Let i be big enough to have f i (a) = f i (b) and let w, y be any edge on the unique arc in G i joining f i (a) and f i (b).Consider the graph H obtained from G i by splitting the edge w, y into two edges w, x i p , x i p , y and by adding p − 2 new vertices z 1 , . . ., z p−2 with edges x i p , z j for all 1 ≤ j ≤ p.Note that there is an epimorphism ) and the identity otherwise, and note that in H there is a point of order p on the arc connecting f i (a) to f i (b).We can now find k big enough so that there is an epimorphism i and note that in G k there is a point of order p on the arc connecting f k (a) and f k (b), namely the point and note that for all m > k, x m p is a point of order p on the unique arc in G m joining f m (a) and f m (b).Since x = (x m p ) m∈N ∈ G P is a coherent sequence it determines a ramification point of order p in |G P | by Theorem 4.11 and Remark 4.12.By construction a and b belong to different connected components of G P \ {x}, and since G P is hereditarily unicoherent this implies that any connected set in G P containing both a and b must contain also x.In particular x ∈ π −1 (X) and so π(x) ∈ X.
We can now analyze what happens in the case ω ∈ P , so that we are looking at the families F P with projective Fraïssé limit F P instead.The situation is more subtle.Proof.The proof is very similar to that of Theorem 4.19.For the remainder of this proof we fix a Fraïssé sequence F i for F P , with epimorphisms f i j : F i → F j .We consider F P as a subspace of F i , and call f i : F P → F i the canonical projections.We already know that |F P | is a dendrite so we need to check that all ramification points of |F P | have order in P , and that for all p ∈ P the set is a weakly coherent point of F P by Theorem 4.11, where π : F P → |F P | is the topological realization.Now the way the family F P is constructed allows two options: either ord(x i ) becomes constant for big enough i, so that (x i ) is actually a coherent point of F P and we are in the same case as in the proof of Theorem 4.19, or ord(x i ) → ∞ as i → ∞, which by Remark 4.12 implies that x is a ramification point of |F P | of infinite order.In either case ord(x) ∈ P .
It remains to show that for every p ∈ P the set of ramification points in If p = ω we can repeat verbatim the argument given in the proof of Theorem 4.19, so we need to show that ramification points of infinite order are arcwise dense in Fix also a strictly increasing sequence (n j ) j∈N of natural numbers, such that n j ∈ P for all j and n j → ∞ as j → ∞.Let w, y be any edge on the unique arc in F i joining f i (a) and f i (b).Consider the graph H obtained from F i by splitting the edge w, y into two edges w, x 1 , x 1 , y and by adding Note that there is an epimorphism and the identity otherwise, and note that in H there is a point of order exactly n 1 on the arc connecting f i (a) to f i (b).We can now find k big enough so that there is an epimorphism i and note that in F k there is a point of order at least n 1 on the arc connecting f k (a) and f k (b), namely the point x 2 ∈ (f k H ) −1 (x 1 ) witnessing the weak coherence of f k H at x 1 .Now we can iterate this construction, by considering the tree H 2 obtained from F k by adding at least n s − n 1 − 2 neighbours to x 2 , with the map f : H 2 → F k sending all of those new vertices to x 2 , where s is big enough to have ord(x 2 ) < n s − n 1 − 2. Then we can find some l > k and an epimorphism f l H : and as above we can find a point x 3 ∈ F l of order at least n s on the arc between f l (a) and f l (b).Repeating this construction we build a weakly coherent sequence (x i ) such that ord(x i ) → ∞ and x i lies on the unique arc in F i between f i (a) and f i (b).The same argument used at the end of the proof of Theorem 4.19 allows us to conclude that the ramification point of infinite order π(x i ) ∈ |F P | belongs to the arc X.
Remark 4.21.In Theorem 4.20 we had to assume that {3, 4, 5, . . ., ω} \ P is infinite, which was used to find a sequence n j of natural numbers with lim n j → ∞ and n j ∈ P , in order to construct weakly coherent points of F P corresponding to ramification points of |F P | of infinite order.When P is cofinite in {3, 4, 5, . . ., ω} the family F P is still a projective Fraïssé family, as we showed above, but the problem is that the order of the points in weakly coherent sequences must stabilize instead of growing unboundedly.Indeed the techniques used in the proof of Theorem 4.20 easily establish that |F P | = W P ′ where P ′ = P \ {ω} if P = {3, 4, 5, . . ., ω} and P ′ = (P \ {ω}) ∪ {a} otherwise, where a is the smallest integer such that for all n > a, n ∈ P .

Fraïssé categories and Projection-Embedding pairs
In this section, we construct all generalized Ważewski dendrites as projective Fraïssé limits by moving to the more general setting of Fraïssé categories developed by Kubiś in [16].Those are categories in which the amalgamation and joint embedding properties hold (when expressed in terms of the appropriate diagrams).As an application we recover a countable dense homogeneity result for End(W P ).We quickly recall some definitions from [16], but stress that the approach followed there is far more general than what we need.In particular, since we are only working with countable collections of finite objects (with finitely many morphisms between any two of them) we can ignore all the issues regarding existence of Fraïssé sequences and uniqueness of Fraïssé limits that arise when looking at the uncountable case.Of particular interest to us is section 6 of [16] in which projection-embedding pairs are introduced as a tool to produce objects that are universal both in an injective and a projective sense.

Fraïssé Categories.
The following definitions and results are taken from [16,, where they are presented in a more general way, which works in the uncountable case as well.
As a warning let us point out that while Kubiś's approach is phrased in terms of injective Fraïssé limits we decided to phrase everything in terms of projective Fraïssé limits.There is no real difference between the two approaches, since it is enough to look at the opposite category K op to translate between the two.
Given a category K and two objects a, b ∈ K, let K(a, b) denote the morphisms a → b in K. Definition 5.1.Let K be a category.We say that K has the amalgamation property if for all a, b, c ∈ K and all morphisms f ∈ K(b, a), g ∈ K(c, a), there exist d ∈ K and morphisms We now wish to define a category of projective sequences in K, which we denote by σK, whose morphisms will be defined in a way that guarantees the following property.Suppose that x, ȳ ∈ σK and K is embedded in a category in which those two sequences have a limit.Given an arrow in σK(x, ȳ) we want an induced arrow lim ← − x → lim ← − ȳ.Given x ∈ σK we denote x(n) by x n and the morphism x(n) → x(m) for n ≥ m by x n m .By definition a sequence in K is a functor ω → K, where we think about the former as a poset category with the reverse order, so a transformation x → ȳ is by definition a natural transformation F : x → ȳ •ϕ, where ϕ : ω → ω is order preserving.To define an arrow x → ȳ in σK we need to identify some of those natural transformations.
Definition 5.2.Let F : x → ȳ • ϕ and G : x → ȳ • ψ be two natural transformations.We say that F and G are equivalent if (1) For every n there exist m ≥ n such that ϕ(m) ≥ ψ(n) and (2) For every n there exist m ≥ n such that ψ(m) ≥ ϕ(n) and n .An arrow x → ȳ in σK is an equivalence class of natural transformations Note that the category σK in Kubiś's approach is the analogue of the class of structures F ω in the approach by Panagiotopolous and Solecki.We can now define a Fraïssé sequence for a category K, which is the analogue of the Fraïssé sequence in the classical or projective Fraïssé limit construction.Kubiś defined in an analogous manner a category of sequences of length κ for any cardinal κ.Similarly the following definition is a special case of the definition of a κ-Fraïssé sequence from [16].Definition 5.3.Let K be a category.A Fraïssé sequence in K is a projective sequence ū : ω → K satisfying: (1) For every x ∈ K there exist n < ω with K(u n , x) = ∅.
(2) For every n < ω and every arrow f ∈ K(y, u n ), with y ∈ K, there exist m ≥ n and g ∈ K(u m , y) such that f • g = u m n .Note that if K has the amalgamation property then ū also satisfies The following is a special case of Corollary 3.8 of [16] and shows existence of Fraïssé sequences: Lemma 5.4.Let K be a category with the amalgamation and joint embedding property with countably many objects and such that between any two objects there are finitely many arrows.Then K has a Fraïssé sequence.
The following is Theorem 3.15 of [16] and shows the uniqueness (up to isomorphism) of Fraïssé sequences, hence also the uniqueness of Fraïssé limits when we embed into a category in which the Fraïssé sequence has a limit: Theorem 5.5.Assume u and v are Fraïssé sequences in a given category K. Assume further that k, l < ω and f ∈ K(u k , v l ).Then there exists an isomorphism In particular u ≈ v.

5.2.
Generalized Ważewski Dendrites from Fraïssé Categories.We now have all the tools needed to talk about projection-embedding categories and construct the generalized Ważewski dendrites not obtained in previous sections.
Definition 5.6.Given P ⊆ {3, 4, . . ., ω} let L ′ P denote the language with a binary relation R and unary relations U p for every p ∈ P .Let L R = {R} ⊆ L ′ P and let L P = {U p | p ∈ P } ⊆ L ′ P .Let ‡F P denote the following category: • A ∈ ‡F P iff A is a finite tree (with R as the edge relation), A contains at least one ramification point and every vertex of A is either an endpoint or a ramification point.Moreover for every ramification point a ∈ A there is exactly one p ∈ P such that U p (a) holds, and in that case ord(a) ≤ p.When a is a ramification point we will refer to the unique p ∈ P for which U p (a) holds as the label of a.If a instead is an endpoint of A, then U p (a) doesn't hold for any p ∈ P .
• Given A, B ∈ ‡F P an arrow f ∈ ‡F P (B, A) is a pair (p(f ), e(f )) where p(f ) : B → A is a weakly coherent epimorphism from B to A viewed as L R -structures with the discrete topology.Moreover if for a ∈ A, U p (a) holds and b ∈ B witnesses the weak coherence of p(f ) at a, then U p (b) holds.In particular, we are not requiring that if U p (b) holds, then U p (p(f )(b)) also holds.On the other hand e(f ) : End(A) → End(B) is a map associating to every a ∈ End(A) , where End(X) is the set of endpoints of X.In other words p(f ) is a weakly coherent epimorphism between the L R -reducts, while e(f ) is a partial right inverse defined on the endpoints.
Given two arrows f : B → A and g : C → B in ‡F P we define their composition in the obvious way: f • g is the pair (p(f ) • p(g), e(g) • e(f )).It is clear that composition is associative and that the pair (Id A , Id A ) is the identity morphism for A in ‡F P , so ‡F P is a category.
Remark 5.7.The argument used to prove Lemma 4.16 shows that ‡F P is directed, so in order to check that it is a Fraïssé category we only need to check that it has the amalgamation property.
Lemma 5.8.The category ‡F P has amalgamation.In other words whenever we have arrows f ∈ ‡F P (B, A), g ∈ ‡F P (C, A), there exists D ∈ ‡F P with arrows Proof.The proof is very similar to the proof of amalgamation given in the previous section for F P , so we don't write all the details.Instead we point out the extra steps that need to be taken to adapt the proof of 4.17 to the current setting.Let A, B, C, f, g be as in the statement of the lemma.We proceed by induction on |E(B)| + |E(C)|.The smallest structures in ‡F P are trees with a single ramification point labelled with any p ∈ P that has 3 neighbours.Clearly if B and C are of that form then A must be the same tree on 4 vertices as well, and all the ramification points in A, B, C must have the same label, so we can take D = A = B = C with identity maps to be the amalgam.Our induction hypothesis now becomes that not only diagrams of the form B → A ← C with smaller |E(B)| + |E(C)| can be amalgamated in ‡F P , but that the amalgamation can be carried on in a way that preserves labels and is compatible with the embedding part of the arrows.With this assumption we can use the same construction as in the proof of Lemma 4.17.Indeed as in that lemma suppose without loss of generality that p(f ) −1 (a) is nontrivial for some a ∈ A. Suppose first that a is a ramification point and we are in the B 2 = B 1 case.We can produce and h ′ 2 ∈ ‡F P (D ′ , C) as in the proof of Lemma 4.17.Note that in this case p(π) is the identity map between End(B) and End(B ′ ), so that e(f ′ ) can be defined to agree with e(f ).We then build D from D ′ by glueing back B 2 along an edge.Points in D are already labeled correctly, with labels coming from either B 2 or D ′ , and since glueing B 2 into D ′ doesn't collapse any endpoints in D ′ we can take e(h 2 ) = e(h ′ 2 ), while e(h 1 ) = e(h ′ 1 ) on End(B ′ ) and is the identity on End(B 2 ) \ {b 1 , b 2 }.
The B 2 = B 1 is easier to deal with: we follow the same construction as in the proof of Lemma 4.17 and we never run into issues with the embedding part of the morphisms, since endpoints are never collapsed during the construction.
If instead a is an endpoint there are still no issues mimicking the construction from Lemma 4.17 as far as labels are concerned, but a little more care is needed for the embedding part of the arrows, so we will write out some of the details.Let b ∈ p(f ) −1 (a) be the unique vertex for which there is a b As in the previous case we construct B ′ from B by collapsing B 2 to b through the map p(π) : B → B ′ but now we also need to define the embedding part e(π) in order to get a morphism in ‡F P .Let e(π) be the identity on End(B ′ )\{b} and e(π)(p(π)(b)) = e(f )(a).Similarly we obtain a coherent epimorphism p(f ′ ) : B ′ → A as the only such morphism satisfying p(f ′ ) • p(π) = p(f ).There is only one choice for e(f ′ ), namely Once again we obtain, by inductive hypothesis, an amalgam We now have to glue back B 2 to D ′ in order to build an amalgam D with maps h 1 ∈ ‡F P (D, B) and h 2 ∈ ‡F P (D, C).We already know how to build D as a graph and how to define p(h 1 ), p(h 2 ), since the construction is the same as the one in the proof of Lemma 4.17.As in the previous section we obtain D from D ′ and B 2 by identifying b ∈ B 2 with an endpoint of D ′ .Here we take α ∈ End(D ′ ) to be e(h ′ 1 )(b) (which is also equal to e(h ′ 2 )(e(g)(a))).Points of D are already correctly labeled, with labels coming from either B 2 or D, and we define p(h 1 ), p(h 2 ) exactly as above.We only need to define e(h 1 ), e(h 2 ).There is nothing to be done for e(h 2 ) which we simply define to be equal to e(h ′ 2 ) on all endpoints except e(g)(a) ∈ B 2 , where it now takes value e(f )(a).To define e(h 1 ) there are a few cases to consider, as follows: It is now easy to check that the resulting square diagram is commutative, as desired.
Since ‡F P has countably many objects and between any two objects of ‡F P there are finitely many arrows, Lemma 5.4 and Theorem 5.5 immediately imply the following: Theorem 5.9.The category ‡F P has a Fraïssé sequence u and any two Fraïssé sequences are isomorphic as elements of σ ‡F P .In particular whenever ‡F P is embedded in a category in which Fraïssé sequences have colimits, the colimits arising from two Fraïssé sequences are isomorphic.
As mentioned in the introduction to this section we want to embed σ ‡F P in a category in which those sequences actually have a limit.We do so by considering σ ‡F P as a subcategory of a category G of labeled topological graphs defined as follows: The objects of G are pairs G = (B, E) where B is a graph-dendrite, equipped with unary predicates U n , for n ∈ P , such that if a is a ramification point of G, there is a unique n for which U n (a) holds and n ≤ ord(a), and E is a countable subset of U. The morphisms of G from (B, E) to (A, F ) are pairs f = (p(f ), e(f )) such that e(f ) : F → E is an injection, while p(f ) : B → A is an epimorphism of topological graphs such that p(f ) • e(f ) = Id F .
To a sequence u ∈ σ ‡F P we associate the pair (U, E), where U is the topological graph obtained as the projective limit of the projection part of u, while E is the direct limit of the embedding part of u.Moreover every ramification point x of U is labeled with its order as label, where the order of x is the (potentially infinite) number of components of U \ {x}.Given another v ∈ σ ‡F P , a morphism ψ : u → v is by definition a natural transformation ψ from u to v • ϕ, where ϕ : ω → ω is order-preserving.It is easy to check that if (U, E) and (V, F ) are the objects associated to u and v as described above, then the projection part of ψ converges to a continuous surjection p(ψ) : U → V, while the embedding part converges to an injection e(ψ) : F → E such that p(ψ) • e(ψ) = Id F .Note that, since in Definition 5.6 we required that the witness of weak coherence at a point a has the same label as a, p(ψ) has the property that if v ∈ V is a ramification point with label l, then there is a ramification point u ∈ p(ψ) −1 (v) with the same label.This describes the desired functor σ ‡F P → G.
To a sequence u ∈ σ ‡F P we associated a pair (U, E), where U is a topological graph.We will show that whenever u is a Fraïssé sequence for ‡F P , then |U| is the generalized Ważewski dendrite W P , where π : U → |U| is the topological realization.
Note that the embedding part of u plays no role in the following theorem.This observation will be made precise and become relevant later.
Theorem 5.10.Let u = F i | i < ω with maps f i+1 i ∈ ‡F P (F i+1 , F i ) be a Fraïssé sequence for ‡F P and let (F P , E) be its limit.Then |F P | = W P .
Proof.Even though ‡F P doesn't allow splitting edges, we can proceed as in Remark 4.15 to use all the results that required splitting edges.Because of this we know that |F P | is a dendrite thanks to Theorem 3.17 since p(f i+1 i ) is monotone and (weakly) coherent, moreover we still have the correspondence between (weakly) coherent points of F P and ramification points of |F P | established in Theorem 4.11 and Theorem 4.13.The fact that each ramification point has order in P follows from an argument similar to the one done in the proof of Theorem 4.20: if x = (x n ) ∈ F P is labelled with U p , then x n must be labelled with U p from some index on.If p < ω, then ord(x n ) must also be p from some (possibly bigger) index on, which implies that ord(x) = p, and so ord(π(x)) = p.If x = (x n ) ∈ F P is labelled with U ω instead, then ord(x n ) → ∞ as n → ∞ as in the proof of Theorem 4.20, hence ord(π(x)) = ω.The fact that the ramification points of each order in P are arcwise dense in |F p | follows from exactly the same argument as the one at the end of the proof of Theorem 4.20.

A Countable Dense Homogeneity Result for End(W P ).
We now want to prove the following homogeneity result for the endpoints of W P , first stated in [5] for W 3 : Theorem 5.11.Let P ⊆ {3, 4, . . ., ω} and let Q 1 , Q 2 be countable dense subsets of End(W P ).Then there is a homeomorphism h : We begin with a more precise description of the endpoints of a projective Fraïssé limit of trees, in particular we want to obtain that if (U, E) is the projective Fraïssé limit of ‡F P , then π(E) is a dense set of endpoints in |U|.We first show that endpoints are edge related only to themselves.Proposition 5.12.Let F be a projective Fraïssé family of trees with (weakly) coherent epimorphisms that allows splitting edges and let F be its projective Fraïssé limit.If e ∈ End(F), then there is no e ′ ∈ F\{e} with e, e ′ ∈ E(F).
Proof.Suppose for a contradiction that there is e = e ′ ∈ F with e, e ′ ∈ E(F).Let F n with maps f m n : F m → F n be a Fraïssé sequence for F and let g n : F → F n be the canonical projections.Let e n , e ′ n be g n (e) and g n (e ′ ) respectively.First we claim that for all but finitely many n, e n ∈ End(F n ).Indeed suppose for a contradiction that this is not the case and let m be big enough so that e m = e ′ m and e m is not in End(F m ).Let a m be a point in F m such that e m is on the unique arc joining a m and e ′ m .Without loss of generality we can assume that a m is a ramification point (we can for example take a m = e m ) and let, for k > m, a k denote the vertex of We have thus obtained that, if e ∈ End(F) and e ′ ∈ F is such that e, e ′ ∈ E(F), then for all but finitely many n ∈ N, e n is an endpoint of F n .Now let H be the tree obtained by splitting the e m , e ′ m edge, so that V (H) = V (F m ) ⊔ {x} and x, e ′ m } and consider the epimorphism ϕ : H → F m defined by ϕ(x) = e m and ϕ is the identity on H otherwise (when applying this result to the F = ‡F P case we will also need to add neighbours to x until it has the correct order).By assumption we can find n > m and an epimorphism ψ : But now, since e n ∈ End(F n ) and ψ is monotone, it is impossible for e m to be in the image of ψ, contradicting that the latter is an epimorphism.Lemma 5.13.Let F be a projective Fraïssé family of trees with (weakly) coherent epimorphisms that allows splitting edges.Let F i | i < ω be a Fraïssé sequence for F and let F ⊆ F i be its projective Fraïssé limit.Let e = (e n ) n∈N ∈ F. If there is N ∈ N such that for every m ≥ N, e m is an endpoint of F m , then e ∈ End(F).
Proof.Assume that (e n ) n∈N is such that e m is an endpoint of F m for all m ≥ N. Suppose by contradiction that e is not an endpoint of F and let A ⊆ F be an arc with e ∈ A, but e = a 1 , a 2 , where a 1 , a 2 are the endpoints of A. Let k be big enough so that e k is an endpoint of F k , and f k (a 1 ), f k (a 2 ), f k (e) are pairwise distinct.By Theorem 3.18 f k is monotone, so by Lemma 3.14 f k (a 1 ), f k (a 2 ) and e k all belong to End(f k (A)), a contradiction.Lemma 5.14.Let ‡F P be as above, let u = F i | i < ω be a Fraïssé sequence for ‡F P with maps f m n ∈ ‡F P (F m , F n ) and let (F, E) be its projective Fraïssé limit.Then E ⊆ End(F) and E is dense in F.
Proof.It follows from Lemma 5.13, that E ⊆ End(F).Let now U ⊆ F be a nonempty open set.We want to show that E ∩ U = ∅.Let n ∈ ω be sufficiently big so that there is x ∈ F n such that f −1 n (x) ⊆ U. Consider any finite tree T ∈ ‡F P with an epimorphism f ∈ ‡F P (T, F n ) such that p(f ) −1 (x) contains an endpoint of T .By definition of a Fraïssé sequence we can find m ≥ n and g ∈ ‡F P (F m , T ) such that f • g = f m n .Since the preimage of an endpoint through a (weakly) coherent map always contains an endpoint, this implies that there is an endpoint y ∈ End(F m ) with f −1 m (y) ⊆ U. Now the point of the limit determined by the sequence e(f k m )(y) for k ≥ m is an endpoint by Lemma 5.13 and is contained in both E and U by construction.
As a consequence of the previous two lemmas and Proposition 5.12, we obtain the following corollary: Corollary 5.15.If (U, E) is the projective Fraïssé limit of ‡F P , then π(E) is a dense set of endpoints in |U| ≃ W P , where π : U → |U| is the topological realization.
The last piece we need before being able to prove the desired countable dense homogeneity result for the endpoints of W P is Theorem 5.22.Intuitively this theorem will say that the embedding parts of the morphisms is not important in determining whether a sequence is Fraïssé for ‡F P .This is not surprising, since in Theorem 5.10 the embedding part of the morphisms played no role.We first need to introduce new definitions and make a few simple remarks.Definition 5.16.Let X be a dendrite.Given any three distinct x, y, z ∈ X, their center C(x, y, z) is the unique point in If X is a graph-dendrite instead we have analogous notions, for any three distinct points x, y, z ∈ X we define their center C(x, y, z) to be the unique point in [x, y] ∩ [y, z] ∩ [x, z], where those are now arcs in the sense of topological graphs.The other notions are defined similarly.Definition 5.17.An immersion of a finite tree A in a graph-dendrite B is a map f : A → B (which will usually not be a graph homomorphism) such that for all distinct a, b, In other words there exist a topological graph A ′ obtained from A by replacing every edge of A with an arc (as in Definition 3.3) and a graph embedding Note that the image of an immersion f : A → B is center-closed, so it determines a partition Ω f = Ω f (A) of B \ f (A) as above.There are two particular instances of immersions that will be relevant.They are described in the following examples.
Example 5.18.An arrow f ∈ ‡F P (B, A) determines an immersion i(f where w(a) ∈ B is the witness for the (weak) coherence of p(f ) at a.
Example 5.19.Let F n | n < ω with maps f m n ∈ ‡F P (F m , F n ) be a sequence in ‡F P and let F with maps f ∞ n ∈ ‡F P (F, F n ) be its limit.We have immersions h n : F n → F given by x → (p(f n 0 )(x), p(f n 1 )(x), . . ., p(f n n−1 )(x), x, w n n+1 (x), w n n+2 (x), . ..),where w i j (x) denotes the witness for the (weak) coherence of f j i : F j → F i at x for x a ramification point.For endpoints we follow the embeddings instead of the witnesses, that is when x is an endpoint.
Remark 5.20.Let i : A → B be an immersion of finite trees.If f : B → A is a monotone map such that • for every a ∈ A, f (i(a)) = a, • for every a ∈ A and every b Then f is weakly coherent.
Definition 5.21.We say that a topological graph G is a W P -prespace if G is a prespace, G is a graph-dendrite, for every p ∈ P the set of ramification points of order p is arcwise dense in G and every ramification point of G has order in P .In particular if G is a W P -prespace, then |G| = W P .
We can now characterize Fraïssé sequences in ‡F P .
Theorem 5.22.A sequence F n | n < ω with maps f m n ∈ ‡F P (F m , F n ) is Fraïssé if and only if the inverse limit of p(f m n ) as a topological graph is a W P -prespace.
Proof.The if part of the statement was already proved in Theorem 5.10.Suppose that is a W P -prespace.We want to show that F n |< ω is a Fraïssé sequence for ‡F P .In other words given A ∈ ‡F P and an epimorphism g ∈ ‡F P (A, F n ), we want to find m ≥ n and an epimorphism h ∈ ‡F P (F m , A) such that As in Example 5.19 we obtain an immersion h n : F n → F. Moreover we also have an immersion j : F n → A as in Example 5.18.Using the fact that F is a W P -prespace we can find an immersion k : A → F such that k • j = h n .We construct k inductively on elements of A. There is only one choice for the value of k on j(F n ) so that k • j = h n .Enumerate A\j(F n ) as {a 1 , . . ., a r }.We define k(a 1 ) to be any point of F with ord(a 1 ) = ord(k(a 1 )) so that k is an isomorphism of trees with the betweenness relation from j(F n ) ∪ {a 1 } to h n (A) ∪ {k(a 1 )}.There exists such a point k(a 1 ) ∈ F because F is a W P -prespace.Proceeding inductively in a similar manner, we define k(a i ) for all i ≤ r.Now we can find m ≥ n such that, calling We can now define a weakly coherent map from F m to A, we only need to be careful and make sure that it makes the appropriate diagram commute.Explicitly we define h ∈ ‡F P (F m , A) as follows.Let x ∈ F m .If x = i(a) for some a ∈ A, then p(h)(x) = a.If x ∈ a i(A) (as defined in Definition 5.16) for some a ∈ A, then p(h)(x) = a.If x ∈ C i(a),i(a ′ ) for some a, a ′ ∈ A with (a, a ′ ) ∈ E(A), then we check whether p(g)(a) = f m n (x) or p(g)(a ′ ) = f m n (x) and define p(h)(x) accordingly.Since the map we just defined is monotone, it must also be weakly coherent by Remark 5.20.For the embedding part of We conclude by proving the following theorem, from which Theorem 5.11 follows immediately by the uniqueness of Fraïssé limits.
Theorem 5.23.Let P ⊆ {3, 4, . . ., ω} and let Q ⊆ End(W P ) be a countable dense set of endpoints.Then ‡F P has a Fraïssé sequence u = F i | i < ω with morphisms f m n ∈ ‡F P (F m , F n ) such that if (U, D) is its Fraïssé limit, then π(D) = Q, where π : U → |U| ∼ = W P is the topological realization.
Proof.Enumerate Q without repetitions as {q 1 , q 2 , q 3 , . ..}.Let A 0 = {q 1 , q 2 , q 3 , c} where c = C(q 1 , q 2 , q 3 ) and, for all 0 < n < ω, let A n be the center closure of {q 1 , . . ., q kn }, where (k n ) n<ω is an increasing sequence chosen so that |A n+1 ∩ (a, a ′ )| ≥ 2, whenever a, a ′ ∈ A n are such that [a, a ′ ] ∩ A n = {a, a ′ }, which is possible since Q is dense.We now build a finite graph F n with a vertex v a for every a ∈ A n , and an edge v a , v a ′ iff [a, a ′ ] ∩ A n = {a, a ′ }.We label every v c ∈ F n that comes from a ramification point c ∈ A n with the order of c in W P .It only remains to define morphisms f n+1 n ∈ ‡F P (F n+1 , F n ).Since we have A n ⊆ A n+1 , we also get an immersion i : F n → F n+1 that maps v a ∈ F n to v a ∈ F n+1 , for every a ∈ A n .The embedding part of f n+1 n is exactly i End(Fn) .For the projection part for all pairs a, a ′ ∈ A n with [a, a ′ ] ∩ A n = {a, a ′ }, we partition C va,v a ′ ∪ {v a , v a ′ } ⊆ F n+1 into two sets H, H ′ such that ) is monotone and satisfies the hypothesis of Remark 5.20, so it must be weakly coherent.
Thanks to Theorem 3.7 and the condition |(a, a ′ ) ∩ A n+1 | ≥ 2 the limit of this sequence is a prespace.Since Q is dense we have that n A n = Br(W P )∪Q.Using this fact it is easy to check that the limit of this sequence is a W P -prespace, so that the sequence is Fraïssé by Theorem 5.22.Calling (U, D) its limit, we now only need to identify (|U|, π(D)) with (W P , Q).To any a ∈ A n+1 \ A n we have associated the point of U, which is a ramification point by Theorem 4.11 if a is a ramification point of A n+1 , and which is an endpoint if a is an endpoint of A n+1 .We define a continuous function h : Br(W P ) ∪ Q → |U| by setting h(a) = π(r a ) for every a ∈ n A n .To verify the continuity of h note that if x n → x is a convergent sequence of points in the domain of h whose limit is also in the domain of h, then for any m ∈ N we can find N ∈ N big enough so that A m doesn't separate x k and x, for any k ≥ N. In other words r x k and r x agree in at least the first m coordinates, which implies that r x k → r x , since m was arbitrary.This shows that the function a → r a is continuous, from which the continuity of h follows immediately, since h is obtained by composition of that function with π, which is continuous.Note that the image of h is dense in |U| (in particular, any ramification point of |U| is in the image), h is injective, and h preserves the betweenness relation.We will now extend h to a continuous function h : W P → |U| and then verify that h is injective and surjective.For where h(U) means h(U ∩ (Br(W P ) ∪ Q)).By the Kuratowski's extension theorem (see Theorem 3.8 and its proof in [15]), in order to extend the continuous function h to a continuous function h : W P → |U|, it suffices to show that osc(h, x) = 0 for every x ∈ End(W P ) \ Q or when x is a regular point.
• In the first case let (x i ) i<ω be a sequence in Br(W P ) with x i+1 ∈ [x i , x] for every i, x i → x and, for every i < ω, let C x i (x) be the component of W P \ {x i } containing x. Then {x} = i<ω C x i (x) and we have i<ω h(C x i (x)) = ∅.Indeed if y ∈ Br(W P ) ∪ Q and w is the unique point on [y, x 0 ] ∩ [y, x] ∩ [x 0 , x], then there is i big enough so that w ∈ C x i (x), so that y ∈ C x i (x) and, since h preserves the betweenness relation, h(y) ∈ C x i (x).Since C x i+1 (x) ⊆ C x i (x) for every i (because x i+1 ∈ [x i , x]), this shows that i<ω h(C x i (x)) = ∅.Therefore by compactness of W P we have diam(h(C x i (x))) → 0 and hence osc(h, x) = 0.
• If ord(x) = 2 let (y i ) i<ω and (z i ) i<ω be two sequences in Br(W P ) with y i+1 ∈ [y i , x], y i → x, z i+1 ∈ [x, z i ], z i → x and x ∈ [y i , z i ] for every i < ω.Letting C y i ,z i (x) denote the component of X \ {y i , z i } containing x, we can argue as in the previous case to obtain that {x} = i<ω C y i ,z i (x) and that i<ω h(C y i ,z i (x)) = ∅.Similarly to the previous case, since C y i+1 ,z i+1 (x) ⊆ C y i ,z i (x) for every i, this shows that osc(h, x) = 0.
As discussed above we can now extend h to a continuous function h : W P → |U|.Note that h preserves the betweenness relation.Indeed, the betweenness relation {(x, y, z) ∈ W 3 P | x ∈ [y, z]} is closed in W 3 P .Therefore if x n → x, y n → y, z n → z, x n ∈ [y n , z n ], x ∈ [y, z], x n , y n , z n ∈ Br(W P ), it holds h(x n ) ∈ [ h(y n ), h(z n )], and by continuity of h, we get h(x) ∈ [ h(y), h(z)].
Since h(W P ) ⊇ h(Br(W P ) ∪ Q) = |U|, h must be surjective.To show injectivity of h take x = y ∈ W P such that none of the x, y is a ramification point.Since ramification points are arcwise dense in W P , pick t ∈ Br(W P ) with t ∈ [x, y].Then h(t) ∈ [ h(x), h(y)] and since none of the h(x), h(y) is a ramification point, we obtain h(t) = h(x), h(y).Therefore h(x) = h(y).
Moreover we have h(Q) = π(D) by construction, since h(Q) = h(Q), which concludes the proof.

Example 3 . 2 .
Let C ⊆ [0, 1] be the standard middle-thirds Cantor set.Consider the topological graph G with V (G) = C and for distinct x, y ∈ C we have x, y ∈ E(G) if and only if x, y are the endpoints of one of the intervals removed from [0, 1] in the construction of C (see the picture below for the first few stages of the construction, with the edges represented by solid black lines).

4. 1 .
Weakly Coherent Epimorphisms.Definition 4.1.Let A, B be finite trees and f : B → A a monotone epimorphism.Fix a ∈ A with ord(a) = n ≥ 3 and enumerate as A 0 , . . ., A n−1 the connected components of A \ {a}.We say that a is a point of weak coherence for f if there exists b ∈ f −1 (a) such that m = ord(b) ≥ n and an injection

Remark 4 . 3 .
Let A, B be inverse limits of trees, f : B → A a monotone epimorphism and a ∈ A a ramification point.There exists at most one b ∈ B witnessing the weak coherence of f at a. Consider now a third tree C with a monotone epimorphism g : C → B. If f • g is weakly coherent at a with witness c, then f is weakly coherent at a. Then b = g(c) witnesses the weak coherence of f at a, and c witnesses the weak coherence of g at b.If f • g and g are coherent, then f is coherent.

dExample 4 . 6 .
The map f defined by f (a 1 ) = f (a 2 ) = a and mapping every other node to the node with the same name is a weakly coherent epimorphism which is not coherent at a. Let A, B be finite trees in which every vertex has degree at most 3. Then any monotone epimorphism f : A → B is coherent.Indeed fix b ∈ B of order 3 and let b 1 ,b 2 and b 3 be its distinct neighbours.Let b

Remark 4 . 7 .
Note that if A, B, C are finite trees and f : A → B, g : B → C are (weakly) coherent epimorphisms, then g • f : A → C is also (weakly) coherent.Indeed if c ∈ C with ord(c) ≥ 3 then there is some b ∈ g −1 (c) witnessing the (weak) coherence of g at c and some a ∈ f −1 (b) witnessing the (weak) coherence of f at b.It is easy to check that a also witnesses the (weak) coherence of g • f at c.

Remark 4 . 8 .
Note also that if A is a finite tree with at least two vertices and B with V (B) = V (A) ⊔ { * } obtained from A by splitting an edge as in Definition 3.8, then the two maps B → A mapping * to either endpoint of the split edge are coherent.
) If a ∈ A is such that ord(a) ∈ P and ord(a) ≥ 3 then f is weakly coherent at a, and if b ∈ f −1 (a) is the witness for the weak coherence of f at a, then ord(b) ∈ P .

B 2
with b, b 1 ∈ E(B) and let b 2 be the only element of B 2 for which there exist a b 3 ∈ B 1 \B 2 with b 2 , b 3 ∈ E(B) (uniqueness of b 1 , b 2 , b 3 follows from B being a tree).Consider now the graph B ′ obtained from B by collapsing
{x ∈ |F P | | ord(x) = p} is arcwise dense in |F P |.The first part is very similar to the argument in the proof of Theorem 4.19: if F k witnessing the (weak) coherence of f k m at a m .By monotonicity of f k m and since e k , e ′ k ∈ E(F k ), we have that for all k > m, e k is on the unique arc joining a k and e′ k .Indeed (f k m ) −1 ([a m , e ′ m ]) is connected by monotonicity of f k m and, since F k is a tree, there is a unique edge between (f k m ) −1 ([a m , e m ]) and (f k m ) −1 ([e m , e ′ m ]) which must be the edge between e k and e ′ k .Since the unique arc froma k to e ′ k goes from (f k m ) −1 ([a m , e m ]) to (f k m ) −1 ([e m , e ′ m ]) it must go through this edge, showing that e k lies on the arc from a k to e ′ k .As in the proof of arcwise density in Theorem 4.19 we obtain a ramification point a ∈ F such that e is contained in the arc [a, e ′ ].Since e = a, as the latter is a ramification point, and e = e ′ by assumption, this contradicts the fact that e is an endpoint of F.
its center closure is the smallest center-closed B ⊆ X with A ⊆ B. Given a finite center-closed F ⊆ X let, for a ∈ F , âF denote the set of connected components of X \{a} that contain no point of F .Given distinct a, b ∈ F let C a (b) be the connected component of X \{a} containing b, C b (a) the connected component of X \{b} containing a, and C a,b = C b (a) ∩ C a (b).The partition associated to F is then If a, b, c ∈ A are such that b ∈ [a, c], then k(b) ∈ [k(a), k(c)] because k is an immersion, and i(b) ∈ [i(a), i(c)] because p(f ∞ m ) is monotone so it must map arcs to arcs.Conversely suppose for a contradiction that i(b) ∈ [i(a), i(c)] but b ∈ [a, c].Let d ∈ [a, c] be the only point of [a, c] which belong to the shortest arc from b to [a, c].If d = a, then a ∈ [b, c], which, arguing as in the previous implication, implies k(a) ∈ [k(b), k(c)] and so i(a) ∈ [i(b), i(c)], a contradiction.If d = c we reach a contradiction in the same manner, which only leaves the case d ∈ (a, c).But in the latter case we have k(d) ∈ [k(a), k(c)] and k(b) is in a component of k(d) that contains neither k(a) nor k(c), since k is an immersion.By assumption p(f ∞ m )(a), p(f ∞ m )(b), p(f ∞ m )(c) and p(f ∞ m )(d)are pairwise distinct, so we must have that i(d) ∈ (i(a), i(c)), since p(f ∞ m ) maps arcs to arcs, but we must also have that i(b) is in a component of i(d) containing neither i(a) nor i(c), a contradiction.

h
consider a ∈ End(A).If a ∈ e(g)(End(F n )) set e(h)(a) = e(f m n )(b), where b ∈ End(F n ) is such that e(g)(b) = a.If instead a is not in the image of e(g), then e(h)(a) can be any endpoint in p(h) −1 (a).Note that p(h) • e(h)(a) = a for any a ∈ End(A) and g • h = f m n .

( 1 )
v a ∈ H, v a ′ ∈ H ′ , (2) H, H ′ are connected, (3) |H|, |H ′ | ≥ 2.Note that the last condition is easily satisfied since we assumed that |(a, a ′ )∩A n+1 | ≥ 2. We then define p(f n+1 n )(H) = v a and p(f n+1 n )(H ′ ) = v a ′ .By construction p(f n+1 n r a = (p(f n+1 0 )(v a ), . . ., p(f n+1 n−1 )(v a ), p(f n+1 n )(v a ), v a , .. ., v a , v a , . ..) x ∈ W P , let osc(h, x) = inf{diam h(U) | U is an open neighbourhood of x}, Definition 3.3.A topological graph G is called disconnected if it is possible to partition G into two nonempty disjoint closed sets A, B such that whenever a ∈ A, b ∈ B, we have a, b ∈ E(G).A topological graph is called connected if it is not disconnected.A topological graph A is called an arc if it is connected,but for all a ∈ A, except at most two vertices, called the endpoints, A \ {a} is disconnected.Example 3.4.The Cantor graph of Example 3.2 is a connected topological graph.In fact it is an arc.Note that while there is a unique topological arc, namely the interval [0, 1], there are many nonisomorphic topological graphs which are arcs.Indeed, there are finite, countable and uncountable topological graphs which are arcs.
as witnessed by p k ∈ H.By definition of Fraïssé sequence we can find l big enough and an epimorphism f l H On one hand we now must have f l H (p l ) = p k ∈ H by weak coherence, since p l witnesses the weak coherence of f l k at p k ∈ G k and p k ∈ H witnesses the weak coherence of f H k at p k ∈ G k , on the other hand we must have f l H (p l ) = p ′ k , since f l H maps edges to edges, a contradiction.