Topological infinite gammoids, and a new Menger-type theorem for infinite graphs

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid. As our main tool, we prove for any infinite graph $G$ with vertex sets $A$ and $B$ that if every finite subset of $A$ is linked to $B$ by disjoint paths, then the whole of $A$ can be linked to the closure of $B$ by disjoint paths or rays in a natural topology on $G$ and its ends. This latter theorem re-proves and strengthens the infinite Menger theorem of Aharoni and Berger for `well-separated' sets $A$ and $B$. It also implies the topological Menger theorem of Diestel for locally finite graphs.


Introduction
Unlike finite gammoids, traditional infinite gammoids do not necessarily define a matroid. Diestel [9] asked whether a suitable topological notion of infinite gammoid might mend this, so that gammoids always give rise to a matroid. We answer this in the positive by developing such a topological notion of infinite gammoid. Our main tool is a new topological variant of Menger's theorem for infinite graphs, which is also interesting in its own right.
Given a directed graph G with a set B ⊆ V (G) of vertices, the set L(G, B) contains all vertex sets I that can be linked by vertex-disjoint directed paths 1 to B. If G is finite, L(G, B) is the set of independent sets of a matroid, called the gammoid of G with respect to B. If G is infinite, L(G, B) does not always define a matroid [1].
In 1968, Perfect [11] looked at the question of when L(G, B) is a matroid. As usual at that time, she restricted her attention to matroids with every circuit finite, now called finitary matroids. In [5], Bruhn et al found a more general notion of infinite matroids, which are closed under duality and need not be finitary. Afzali, Law and Müller [1] study infinite gammoids in this more general setting and find conditions under which L(G, B) is a matroid. In this paper, we introduce a topological notion of gammoids in infinite graphs that always define a matroid.
These gammoids can be defined formally without any reference to topology, as follows. A ray R in G dominates B if G contains infinitely many vertex-disjoint directed paths from R to B. A vertex v dominates B if there are infinitely many directed paths from v to B that are vertex-disjoint except in v. A path dominates B if its last vertex dominates B. A domination linkage from A to B is a family of vertex-disjoint directed paths or rays (Q a | a ∈ A) where Q a starts in a and either ends in some vertex of B or else dominates B. A vertex set I is in L T (G, B) if there is a domination linkage from I to B. We offer the following solution to Diestel's question: When G is undirected 2 , Theorem 1.1 has the following topological interpretation. On G and its ends consider the topology whose basic open sets are the components C of G \ X where X is a finite set of inner points of edges, together with the ends that have rays in C. The closure of B ⊆ V (G) consists of B, the vertices dominating B, and the ends ω whose rays R ∈ ω dominate B. Thus I ∈ L T (G, B) if and only if the whole of I can be linked to the closure of B by vertex-disjoint paths or rays. 3 We shall not need this topological interpretation: Theorem 1.1 can be used to prove that under certain conditions the naive, non-topological, gammoid L(G, B) is a matroid, too: Corollary 1.2. Let G be a digraph with a set B of vertices such that there are neither infinitely many vertex-disjoint rays dominating B nor infinitely many vertices dominating B. Then L(G, B) is a matroid. Corollary 1.2 does not follow from the existence criterion of Afzali, Law and Müller for non-topological gammoids. Also its converse is not true, see Section 5 for details.
The main tool in our proof of Theorem 1.1 is a purely graph-theoretic Menger-type theorem, which seems to be interesting in its own right. It is not difficult to show that if there is a domination linkage from A to B, then there is a linkage from every finite subset of A to B. Our theorem says that the converse is also true: (ii) In any infinite undirected graph G, a set A of vertices can be linked by disjoint paths and rays to the closure of another vertex set B if and only if every finite subset can be linked to B by vertex-disjoint paths.
We remark that the proof of Theorem 2.1 is non-trivial and not merely a compactness result. Applying compactness, one would get a topological linkage from A to the closure of B by arbitrary topological arcs, not necessarily paths and rays. Our graph-theoretical version of Theorem 2.1 is considerably stronger than this purely topological variant.
In Section 4 we study the relationship between Theorem 1.3 and existing Menger-type theorems for infinite graphs: the Aharoni-Berger theorem [3] and the topological Menger theorem for arbitrary infinite graphs. The latter was proved by Bruhn, Diestel and Stein [6], extending an earlier result of Diestel [8] for countable graphs. For infinite graphs with 'well-separated' sets A and B (defined in Section 4), Theorem 1.3 implies and strengthens the Aharoni-Berger theorem. This in turn allows us to give a proof of the topological Menger theorem for locally finite graphs which, unlike the earlier proofs, does not rely on the (countable) Aharoni-Berger theorem (which was proved earlier by Aharoni [2]).
The paper is organised as follows. After a short preliminary section we prove in Section 3 the directed edge version of Theorem 1.3. In Section 4 we sketch how this variant implies Theorem 1.3, and how Theorem 1.3 implies the Aharoni-Berger theorem for 'well-separated' sets A and B, and the topological Menger theorem for locally finite graphs. In Section 5 we summarise some basics about infinite matroids, and prove Theorem 1.1 and Corollary 1.2.

Preliminaries
Throughout, notation and terminology for graphs are that of [7]. In this paper, we will mainly be concerned with sets of edge-disjoint directed paths. Thus, we abbreviate edge-disjoint by disjoint, edge-separator by separator and directed path by path. Given a digraph G and A, B ⊆ V (G), a linkage from A to B is a set of disjoint paths from the whole of A to B. We update the definitions of when a ray dominates B, a vertex dominates B, a path dominates B, and what a domination linkage is: these are the definitions made in the Introduction with "vertex-disjoint" replaced by "edge-disjoint". The proof of the following theorem takes the whole of Section 3. We delay the proof that Theorem 2.1 implies Theorem 1.3 until Section 4. In a slight abuse of notion, we shall suppress the set brackets of {b} and just talk about "domination linkages from I to b". One implication of Theorem 2.1 is indeed easy: If there is a domination linkage from I to b, then every finite subset S of I has a linkage into b.
Proof. For s ∈ S, let P s be the path or ray from the domination linkage starting in s. Suppose for a contradiction, there is no linkage from S into b. Then by Menger's theorem, there is a set F of at most |S| − 1 edges such that after its removal there is no directed path from S to b.
Suppose for a contradiction that there is some P s not containing an edge of F . Then P s cannot end at b. So P s dominates b, and thus there is some P s -b-path avoiding F , contradicting the fact that F was an edge-separator. Thus each P s contains an edge of F . As the P s are edge-disjoint, |F | ≥ |S|, which is the desired contradiction.

Proof of Theorem 2.1
The proof of Theorem 2.1 takes the whole of this section.

Exact graphs
The core of the proof of Theorem 2.1 is the special case where G is exact (defined below). In this subsection, we show that the special case of Theo-rem 2.1 where G is exact implies the general theorem. More precisely, we prove that the Lemma 3.1 below implies Theorem 2.1.
Given a vertex set D, an edge is D-crossing (or crossing for D) if its starting vertex is in D and the endvertex is outside. We abbreviate V (G)\D by D . The order of D is the number of D-crossing edges. The vertex set D is exact (for some set I ⊆ V (G) and b ∈ V (G)) if b / ∈ D and the order of D is finite and equal to |D ∩ I|. A graph is exact (for b and I) if for every v ∈ V (G) − b, there is an exact set D containing v. First we need some preparation. Let G be a graph and let b ∈ V (G). Let I be the set of all sets I ⊆ V (G) − b such that every finite subset of I has a linkage into b. The following is an easy consequence of Zorn's lemma.
Remark 3.2. Let I ∈ I, and X ⊆ E(G), then there is J ∈ I maximal with I ⊆ J ⊆ X. Proof. By the maximality of I, there is a finite subset I of I such that I + v cannot be linked to b. By Menger's theorem, there is a set D of order at most |I | not containing b but containing I + v ⊆ D. The order must be precisely |I | since I can be linked to b. Thus D is exact, which completes the proof.
Proof that Lemma 3.1 implies Theorem 2.1. By Lemma 2.2, it suffices to prove the "if"-implication. Let G, b, I be as in Theorem 2.1. We obtain the graph H 1 from G by identifying b with all vertices v such that there are infinitely many edge-disjoint v-b-paths. Note that in H 1 every vertex v = b can be separated from b by a finite separator.
It suffices to prove the theorem for H 1 since then the set of dominating paths and rays we get for H 1 extends to a set of dominating paths and rays for G by adding a singleton path for every vertex in I that is identified with b in H 1 .
We build an exact graph H 2 that has H 1 as a subgraph. Let v ∈ V (H 1 )− b. Let k v be the smallest order of some vertex set D containing D and not containing b. By construction of H 1 , the number k v is finite.
We obtain H 2 from H 1 by for each v ∈ V (H 1 ) − b adding k v -many vertices whose forward neighbourhood is that of v and that do not have any incoming edges. We shall refer to these newly added vertices for the vertex v as the clones of v. Now we extend I to a maximal set I 2 ⊆ V (H 2 ) − b such that every finite subset of I 2 has a linkage into b. This is possible by Remark 3.2.
Next we show that H 2 is exact with respect to I 2 . Suppose for a contradiction that there is some v ∈ I 2 such that there is no exact D containing v. First we consider the case that v ∈ V (H 1 ). Since v together with all its clones cannot be linked to b, there is a clone w of v that is not in I 2 . Since w ∈ I 2 , there must be some exact D containing w by Lemma 3.3. If v ∈ D , we are done, otherwise we get a contradiction since there is no linkage from Having shown that H 2 is exact, we now use the assumption that Lemma 3.1 is true for H 2 and I 2 : We get for each v ∈ I some path or ray that dominates b in H 2 . This path or ray also dominates in H 1 because a clone-vertex cannot be an interior vertex of any directed path or ray. And it also dominates in G, which completes the proof that Lemma 3.1 implies Theorem 2.1.

Exact vertex sets
In this subsection, we prove some lemmas needed in the proof of Lemma 3.1.
Until the end of the proof of Theorem 2.1, we shall fix a graph G that is exact with respect to a fixed vertex b and some set I ⊆ V (G) − b. We further assume that every finite subset of I has a linkage into b. First we shall prove some lemmas about exact vertex sets. Lemma 3.4. Let D be exact and let P 1 , . . . P n be a linkage from I ∩ D to b. Then each P i contains precisely one D-crossing edge, and each D-crossing edge is contained in one P i .
Proof. Clearly, each P i contains an D-crossing edge. Since the P i are edgedisjoint no two of them contain the same crossing edge. Since D is exact, there are precisely n D-crossing edges, and thus there is precisely one on each P i . Proof. Since D is exact, each D -crossing edge lies on some path of L. On the other hand |I ∩ D | of the paths start in D , and thus contain an Dcrossing edge. So P cannot contain any D -crossing edge. If P meets D , then it would meet D in a last vertex, and the edge pointing away from this vertex would be an D -crossing edge. Hence P does not meet D , which completes the proof. By equations (3) and (2), we get that Combining this with the two inequalities before, we must have that Now it must be that we must have equality in (3). So there cannot be an edge from D \ D to D \ D, which proves (III). This completes the proof.
Lemma 3.7. Let G be an exact graph, and F be a finite set of vertices.  The hullD of a vertex set D consists of those vertices that are separated by the D-crossing edges from b. Note that D ⊆D and that D consists of those vertices v such that there is a v-b-path all of whose internal vertices are outside D. Since every vertex on such a path is inD , the hull of any hullD isD itself.
We say that two vertex sets D and D are equivalent if they have the same hull. This clearly defines an equivalence relation, which we shall call ∼. Note that D ∼ D if and only if D and D have the same crossing edges.
Proof. Clearly, the set of (F ∪F )-crossing edges is equal to the set of (F ∪F )crossing edges, which gives the desired result.
Let P be the path from that linkage that starts in v. By Lemma 3.5, the path P avoids D. So P witnesses that v / ∈D. This is a contradiction, thus I ∩D = I ∩ D.

Good functions
We define what a good function is and prove that the existence of a good function in every exact graph implies Lemma 3.1. First, we fix some notation. Let E be the set of exact vertex sets D, and letL be the set of linkages from finite subsets of I to b. For a vertex set D, the set N (D) consists of D together with all endvertices of D-crossing edges.
For v ∈ I and some linkage L, let Q v (L) denote the path in L starting from v. For every exact D, the edges of Q v (L) contained in G[N (D)] are the edges of some initial path of Q v (L). We call this initial path P v (D; L). We follow the convention that A function f : E →L is good if it satisfies the following: Here the union ranges over all exact F .
Before proving that there is a good function, we first show how to deduce Theorem 2.1 from that. Let us abbreviate P v (D; f (D)) by P v (D; f ). If it is clear by the context, which function f we mean, we even just write P v (D).
Lemma 3.11. Let f : E →L be a partial function satisfying (i) and (ii). Further assume that for any two exact F and F with F ⊆ F and F ∈ dom(f ), also F ∈ dom(f ).
Let v ∈ I, and let D, Proof. D ∩ D is exact by Lemma 3.6 and in the domain of f . Since f satisfies (ii), we get that P v (D ∩ D ) is a subpath of both P v (D) and P v (D ). Let e be the last edge of P v (D ∩ D ), and x be its endpoint in D ∪ D . Now we distinguish three cases. If x ∈ D ∩ D , the edge e is crossing for both D and D , and thus is the last edge of both P v (D) and P v (D ). So , so the lemma is true in this case.
The case x ∈ D ∩ D is similar to the last case. This completes the proof.
For the remainder of this subsection, let us fix a good function f . The last Lemma motivates the following definition. For v ∈ I, let P v be the union of all the paths P v (D) over all exact D containing v. By the last Lemma P v is either a path or a ray. Lemma 3.12. If P v and P w share an edge, then v = w.
Proof. Let e be an edge in both P v and P w .
share the edge e, we must have that v = w, which completes the proof. Proof. Suppose for a contradiction that P v does not end at b. Then P v does not contain b.
Then by Lemma 3.7, there is an exact D with P v ⊆ D. Then P v (D) contains some D-crossing edge whose endvertex does not lie on P v , which gives a contradiction to the construction of P v .
The following lemma tells us that to prove Theorem 2.1, it remains to show that every exact graph has a good function.
Lemma 3.14. Let G be an exact digraph that has a good function. Let b ∈ V (G). Let I ⊆ V (G) − b such that every finite subset of I has a linkage into b. Then there is a domination linkage from I to b.
Proof. Each P v dominates b: If P v is a path, this is shown in Lemma 3.13. If P v is a ray, this follows from the fact that f is good. By Lemma 3.12 all the P v are edge-disjoint, which completes the proof.

Intermezzo: The countable case
The purpose of this subsection is to prove that there is a good function under the assumption that G is countable. This case is easier than the general case and some of the ideas can already be seen in this special case. However, in the general case we do not rely on the countable case. At the end of this subsection, we tell why this proof does not extend to the general case. We think that this helps to get a better understanding of the general case.
Then there is a sequence of exact hulls D n and linkages L n from I ∩D n to b satisfying the following.
Proof. Assume that for all i ≤ n, we already constructed exact hulls D i and linkages L i satisfying 1-4.
Next, we define D n+1 . By Lemma 3.7, there is an exact F n containing v n+1 . By Lemma 3.6, D n ∪ F n is exact. Let D n+1 be a forwarder of D n ∪ F n with respect to the linkage L n , which exists by Lemma 3.8. Let D n+1 be the hull of D n+1 , which is exact by Lemma 3.10.
It remains to construct L n+1 so as to make 3 true. Let L be some linkage from I ∩ D n+1 to b. By Lemma 3.4, for each D n -crossing edge e there is precisely one P e ∈ L n that contains e, and precisely one Q e ∈ L that contains e. Let R e = P e eQ e b. Since P e e ⊆ D n + e and eQ e b ⊆ D n + e, the R e are edge-disjoint. For L n+1 we pick the set of the R e together with all Q ∈ L that do not contain any D n -crossing edge. Clearly L n+1 is an linkage from I ∩ D n+1 to b. And 3 is true by construction. This completes the construction. Having defined f , it remains to check that it is good: clearly it satisfies (i) and (ii), and it just remains to verify (iii). So assume that for some v ∈ I, the union R = F ∈E P v (F ; f (F )) is a ray. Then R = n∈N P v (D n ; L n ). Let e n v be the unique D n -crossing edge on Q v (L n ). Since D n+1 is a forwarder of D n , the path R n v = e n v Q v (L n ) is contained in D n + b and avoids D n+1 . Thus the paths R n v are edge-disjoint and witness that R dominates b. So f is good, which completes the proof.
Remark 3.17. Our proof above heavily relies on the fact that we can find a nested set of exact vertex sets D n indexed with the natural numbers that exhaust the graph (compare 2 in Lemma 3.15). However if we can find such a nested set, then I must be countable since each D n contains only finitely many vertices of I. Thus this proof does not extend to the general case.

Infinite sequences of exact vertex sets
The purpose of this subsection is to prove some lemmas that help proving that there is a good function in every exact graph. These lemmas are about infinite sequences of exact vertex sets. Proof. Suppose for a contradiction that there is a such sequence (D n |n ∈ N). By taking a subsequence if necessary, we may assume that all D n have the same order. Since any two D n are exact and have the same order, we must have I ∩ D 1 = I ∩ D n for every n. Let L be some linkage from (D 1 ∩ I) to b. Any P ∈ L contains a unique D n -crossing edge for every n by Lemma 3.4. Since D n ⊆ D n+1 , there is a large number n P such that for all n ≥ n P it is the same crossing edge. Let m be the maximum of the numbers n P over all P ∈ L. Then for all n ≥ m, the D n have the same crossing edges and thus are equivalent. This is a contradiction, completing the proof.
Lemma 3.19. Let D be exact and let X be a nonempty set of exact D ⊆ D that is closed under ∼ and taking unions. Then there is some D ∈ X including all D ∈ X .
Proof. Suppose for a contradiction that there is no such D ∈ X . We shall construct an infinite sequence (D n |n ∈ N) as in Lemma 3.18.
Let D 1 ∈ X be arbitrary. Since X is ∼-closed, we may assume that D 1 is its hull. Now assume that D n is already constructed. By assumption, there is D n ∈ X with D n ⊆ D n . Let D n = D n ∪ D n . Let D n+1 be the hull of D n . Then D n+1 ∈ X , and D n D n+1 . This completes the construction of the infinite sequence (D n |n ∈ N), which contradicts Lemma 3.18 and hence completes the proof.

Existence of good functions
The purpose of this subsection is to prove that every exact graph has a good function, which implies Theorem 2.1 by Lemma 3.14. Next we shall define when a partial function is good for the following reason. In order to construct a good function f defined on the whole of E we shall construct an ordinal indexed family of good partial functions f α such that if α > β, then the domain of f α includes that of f β and agrees with f β on the domain of f β . Eventually some f α will be defined on the whole of E and will be the desired good function.
The domain of a partial function f is denoted by dom(f ). A partial function f : E →L is good if it satisfies the following: is a ray, then it dominates b. Here the union ranges over all F ∈ dom(f ).
(iv) Let F and F be exact with F ⊆ F . If F is in the domain of f , then so is F .
Note that if F, F ∈ dom(f ), then so is F ∩ F by (iv). Note that each good partial function defined on the whole of E is a good function.
Lemma 3.20. Let f be a partial function with domain X that satisfies (i)-(v). Then there is a good partial functionf whose domain is the ∼-closurê X of X such thatf X = f .
Proof. For each F ∈X, there is someF ∈ X such that F ∼F . We let f (F ) = f (F ). Clearly,f satisfies (i), (iii) and (vi). Since f satisfies (v) and by Remark 3.9,f satisfies (v).
To see thatf satisfies (ii), let v ∈ I and F, F ∈ dom(f ) with F ⊆ F .
To see thatf satisfies (iv), let F and F be exact with F ⊆ F and F ∈X. Then F ∩F is exact, and since f satisfies (iv), it must be in X. Since F ∩F and F have the same crossing edges, they are equivalent. So F ∈X. Sof satisfies (iv). This completes the proof.
For S ⊆ E, let S(iv) ⊆ E denote the smallest set including S that satisfies (iv). Similarly, let S(v) ⊆ E denote the smallest set including S that satisfies (v). Proof. First let D ∈ [S(iv)](v). Then there are F 1 , F 2 ∈ S(iv) such that Then F i ∩ D ∈ S(iv) for i = 1, 2 by (II) of Lemma 3.6. Since D ⊆ F 1 ∪ F 2 , we deduce that D ∈ [S(iv)](v). This completes the proof.
Let X ⊆ E, and D be exact. Then X[D] denotes the smallest set including X + D that satisfies (iv) and (v).

Extending good partial functions
The aim of this subsubsection is to prove the following lemma that helps us building a good function in that it allows us to extend a good partial function a little bit. If D ∈ dom(f ), then we just take g = f . So we may assume that D ∈ dom(f ). Before we define g, we define auxiliary functions g 1 , g 2 and g 3 with domains X 1 , X 2 and X 3 , respectively, such that dom(f ) ⊆ X 1 ⊆ X 2 ⊆ X 3 ⊆ dom(g), and g will be defined such that g X 1 = g 1 , g X 2 = g 2 , and g X 3 = g 3 . We let X 1 = dom(f ) + D.
For all D ∈ dom(f ), we let g 1 (D ) = f (D ). Next we define g 1 (D). Since I ∩ D is finite, there is some linkage from I ∩ D into b. Let P 1 , P 2 , . . . P n be such a linkage. By Lemma 3.19, there is some Since D is exact, each D -crossing edge lies on one of the P i by Lemma 3.4.
We define g 1 (D) as follows. If no D -crossing edge lies on P i , then we put P i into g 1 (D). If some D -crossing edge, say e i , lies on P i , we take the path Q i from the linkage f (D ) that contains e i , and put the path Q i e i P i into g 1 (D). This completes the definition of g 1 (D), and so of g 1 . Proof. Let F, F ∈ X 1 with F ⊆ F . If F is not D, then P v (F ; f (F )) = P v (F ; f (F )) since f satisfies (ii) and (iv).
So we may assume that F = D. Then F ⊆ D ⊆ D. So P v (F ; f (F )) = P v (F ; f (D )) = P v (F ; f (D)), which completes the proof.
Having defined X 1 and g 1 , we now define X 2 and g 2 . We let X 2 = X 1 (iv). For each F ∈ X 2 there is some F ∈ X 1 such that F ⊆ F . We let g 2 (F ) to consists of those paths from g 1 (F ) that start in F . By construction, g 2 satisfies (i) and (iv). By Lemma 3.6(III), P v (F, g 2 (F )) = P v (F, g 1 (F )) for all v ∈ I. Thus g 2 satisfies (ii) as g 1 does.
Having defined g 2 , we now define g 3 . We let X 3 be X 2 (v), which is equal to dom(f )[D]. We let P v = P v (F ; g 2 ) ∪ P v (F ; g 2 ). By Lemma 3.11, it must be that P v = P v (F ; g 2 ) or P v = P v (F ; g 2 ). Since g 2 satisfies (ii), no vertex of P v (F ; g 2 ) that is not on P v (F ; g 2 ) can be in F ∩ F . By (III) of Lemma 3.6, it must be that every vertex of P v (F ; g 2 ) that is not on P v (F ; g 2 ) is in F \F . Hence the P v are edge-disjoint.
By (III) of Lemma 3.6, each P v contains some (F ∪ F )-crossing edge e v . Let L be some linkage from I ∩ (F ∪ F ) to b. Let Q v be the path in L that contains e v . We define g 3 (F ∪ F ) to consist of the paths P v e v Q v . Clearly, By Lemma 3.21, g 3 satisfies not only (v) but also (iv).
Proof. Let v ∈ I and F, F ∈ dom(f ) with F ⊆ F . Our aim is to prove that P v (F ; g 3 (F )) = P v (F ; g 3 (F )). In the definition of g 3 at F , we have picked F 1 and F 2 in X 2 such that F = F 1 ∪ F 2 in order to define g 3 (F ). Similarly, we have picked F 1 and F 2 in X 2 such that with F = F 1 ∪ F 2 to define g 3 (F ). It suffices to show that P v (X ij ; By the definition of g 3 , we get the following two equations. Since g 2 satisfies (ii), these two equations give the desired result when restricted to X ij . Proof. For each v ∈ I, we compare the sets P v (F ; f (F )) where first the union ranges over all F ∈ dom(f ) and second it ranges over all F ∈ dom(g 3 ).
The second set is a superset of the first and all its additional elements are in P v (D, g 3 ), which is finite. In particular, if the second set is a ray, then so is the first set by Lemma 3.11. In this case, the first set dominates b since f satisfies (iii), so the second set also dominates b. This completes the proof.
Having defined g 3 , we let g =ĝ 3 as in Lemma 3.20. Since g 3 satisfies (i) -(v), g is good by Lemma 3.20. This completes the proof of Lemma 3.22.

Construction of a good function
In this subsubsection, we construct a good function in every exact graph, which is the last step in the proof of Theorem 2.1. Each ordinal α has a unique representation α = β + n where β is the largest limit ordinal smaller than α, and n is a natural number. We say that α is odd if n is odd.
Otherwise it is even. Proof. In order to construct f we shall construct an ordinal indexed family of good partial functions f α such that if α > β, then the domain of f α includes that of f β and agrees with f β on the domain of f β . Eventually some f α will be defined on the whole of E and will be the desired good function. Assume that f β is already defined for all β < α. First we consider the case that α = β + 1 is a successor ordinal. If f β is defined on the whole of E, we stop. Otherwise, we shall find some exact F α . Then we let f α be the partial function g given to us from Lemma 3.22 applied to f β and F α .
How we find F α depends on whether α is an odd or an even successor ordinal. If α is odd, then we pick some D ∈ E \ dom(f β ), and let F α = D.
If α is even, say α = δ + 2n, where δ is the largest limit ordinal less than α, then for F α we pick the forwarder of F δ+n with respect to the linkage f (F δ+n ), which exists by Lemma 3.8.
Having considered the case where α is a successor ordinal, we now consider the case where α is a limit ordinal. For the domain of f α we take the union of the domains of all f β with β < α, and we let f α (D) = f β (D) for some β where this is defined. It is clear that f α satisfies (i),(ii),(iv),(v),(vi), so it remains to show that f satisfies (iii). So let v ∈ I such that R = P v (D; f α ) is a ray. Here the union ranges over all D in the domain of f α . Let O be the set of ordinals β < α such that there is some D ∈ dom(f β ) with v ∈ D. O is nonempty, so it must contain a smallest ordinal . Note that is a successor ordinal. Let − be such that = − + 1.
We shall prove that v ∈ F . Suppose not for a contradiction, We have a representation = δ + k where δ is the largest limit ordinal less than . By construction, the F (l) with (l) = δ + 2 l · k are nested with each other. To prove that R dominates b, it will suffice just to investigate the F (l) .
The paths P v (l) = Q v (f α (F (l) )) are contained in F (l+1) +b. By Lemma 3.4, there is a unique F (l) -crossing edge a l on P v (l). The paths a l P v (l) meet F (l) only in their starting vertex. Thus the paths a l P v (l) are edge disjoint. Since a l is on P v (F (l) ; f α ), it is on R. Hence the paths a l P v (l) witness that R dominates b. So f α is good.
There must be some successor step α = β + 1 at which we stop. Then f β is a good function defined on the whole of E. This completes the proof.
Proof of Theorem 2.1. Recall that the easy implication is already proved in Lemma 2.2. For the other implication, combine Lemma 3.26 with Lemma 3.14 to get a proof of Lemma 3.1. Then remember that Lemma 3.1 implies Theorem 2.1.

Graph-theoretic applications of Theorem 2.1
In this section, we show how Theorem 2.1 implies Theorem 1.3 and how Theorem 1.3 implies the Aharoni-Berger theorem for 'well-separated' sets A and B, and the topological Menger theorem for locally finite graphs.

Variants of Theorem 2.1
In this subsection, we explain how Theorem 2.1 implies Theorem 1.3. Theorem 2.1 is equivalent to the following. Starting with the sketch, one first shows that the directed-edge-version implies the directed-vertex-version for every graph G. For this one considers the auxiliary digraph H of G with V (H) = V (G) × {in, out}. The edges of H are of two types: For each v ∈ V (G), we add an edge pointing from (v, in) to (v, out). For each edge of H pointing from v to w, we add an edge pointing from (v, out) to (w, in). Then the directed-vertex-version for G is equivalent to the directed-edge-version for H.
Next one shows that the directed-vertex-version implies the undirectedvertex-version for every graph G. For this, one considers the directed graph H obtained from G by replacing each edge by two edges in parallel pointing in different directions. Then the undirected-vertex-version for G is equivalent to the directed-vertex-version for H.
Finally, one shows that the undirected-vertex-version implies the undirectededge-version for every graph G. For this, one considers the line graph H of G. Then the undirected-edge-version for G is equivalent to the undirectedvertex-version for H.
It is clear that the directed vertex version of Theorem 4.1 is just Theorem 1.3(i). We call domination linkages in the undirected vertex version vertex-domination linkages. Similarly, we define vertex-linkages. The undirected vertex version of Theorem 4.1 is the following.

Well-separatedness
In this subsection, we prove Corollary 4.3 below, which is used to deduce the Aharoni-Berger theorem for 'well-separated' sets A and B, and the topological Menger theorem for locally finite graphs.
A pair (A, B) of vertex sets is well-separated if every vertex or end can be separated from one of A or B by removing finitely many vertices. Our next aim is to deduce Corollary 4.3 from Corollary 4.2. First we need some lemmas. For this, we fix a graph G and a well-separated pair (A, B) of vertex sets. Let (P a |a ∈ A) be a vertex-domination linkage from A to B. Let ω be an end that cannot be separated from B by removing finitely many vertices. Let A ω be the set of those a ∈ A such that P a is a ray and belongs to ω.

Lemma 4.4.
There is a vertex-linkage (Q a |a ∈ A ω ) from A ω to B such that Q a and P x are vertex-disjoint for all a ∈ A ω and all x ∈ A \ A ω .
Proof. For a finite vertex set S, we denote by C(S, ω) the component of G \ S that contains ω.
As (A, B) is well-separated, there is a finite set of vertices S that separates ω from A. So the set Z of those a ∈ A such that P a meets C(S, ω) ∪ S is finite. As A ω ⊆ Z, the set A ω must be finite. Furthermore there is a finite set T such that C(T, ω) meets precisely those P a with a ∈ A ω . For a ∈ A ω , let t a be first vertex on P a such that t a P a is contained in C(T, ω), which exist as these P a are eventually contained in C(T, ω). The (t a P a |a ∈ A ω ) form a vertex-domination linkage from (t a |a ∈ A ω ) to B in G , where G is obtained from G[C(T, ω)] by deleting all edges on the paths P a t a with a ∈ A ω . By the easy implication of Corollary 4.2 applied to G , we get a vertex-linkage (K a |a ∈ A ω ) from (t a |a ∈ A ω ) to B. Each walk P a t a K a includes a path Q a from a to B. From the construction, it is clear, that the Q a form a vertex-linkage from A ω to B and that Q a and P x are vertex-disjoint for all a ∈ A ω and all x ∈ A \ A ω . Lemma 4.5. There is a vertex-domination linkage (R a |a ∈ A) from A to B such that each R a is a path.
Proof. We shall construct (R a |a ∈ A) by transfinite recursion. First we wellorder the set Ω of ends: Ω = {ω α |α ∈ κ} for κ = |Ω|. At each step β we have a current set of vertex-disjoint A-B-paths Q β . The set A β of start vertices of paths in Q β consists of those a ∈ A such that P a is a ray and belongs to some end ω α with α < β. We shall also ensure that R β = Q β ∪ {P a |a ∈ A β } is a vertex-domination linkage from A to B.
If β is a limit ordinal, we just set Q β = α<β Q α . It is immediate that Q β has the desired property assuming that the Q α with α < β have the property. If β = α + 1 is a successor ordinal, we apply Lemma 4.4 to the vertex-domination linkage R α . Then we let Q α+1 = Q α ∪ {Q a |a ∈ A ω α+1 }. It is clear from that lemma that Q α+1 has the desired property.
This completes the recursive construction. It is clear that Proof that Corollary 4.2 implies Corollary 4.3. Let (A, B) be well-separated such that from every finite subset of A there is a vertex-linkage to B. By Corollary 4.2, there is a vertex-domination linkage (P a |a ∈ A) from the whole of A to B. By Lemma 4.5, we may assume that each P a is a path. However, (P a |a ∈ A) may still contain a path P u that does not end in B. Then P u has to contain a vertex ω that cannot be separated from B by removing finitely many vertices. An argument as in the proof of Lemma 4.4, shows that there is a path Q u from u to some vertex in B such that (P a |a ∈ A − u) together with Q u is a vertex-domination linkage from A to B. Similar as in the proof of Lemma 4.5, we can now apply transfinite induction to replace each P u one by one by such a path Q u . The final vertex-domination linkage is then a vertex-linkage, which completes the proof.

Existing Menger-type theorems
In this subsection, we show how Corollary 4.3 implies the Aharoni-Berger theorem for 'well-separated' sets A and B, and the topological Menger theorem for locally finite graphs.
The Aharoni-Berger theorem [3] says that for every graph G with vertex sets A and B, there is a set of vertex-disjoint A-B-paths together with an A-B-separator consisting of precisely one vertex from each of these paths.
At first glance, it might seem that the Aharoni-Berger theorem does not tell under which conditions there is a linkage from A to B -but actually it does. To explain this, we need a definition. A wave is a set of vertex-disjoint paths from a subset of A to some A-B-separator C. It is not difficult to show that the Aharoni-Berger theorem is equivalent to the following: The whole of A can be linked to B if and only if for every wave there is a linkage from A to its separator set C. Thus Corollary 4.3 implies the Aharoni-Berger theorem for well-separated sets A and B. We remark that neither Corollary 4.3 nor Theorem 2.1 follows from the Aharoni-Berger theorem.
Using this implication, we get the first proof of the topological Menger-Theorem of Diestel [8] for locally finite graphs that does not rely on the Aharoni-Berger theorem. Indeed, the argument of Diestel only relies on the Aharoni-Berger theorem for vertex sets A and B that have disjoint closure in |G|, which is equivalent to being well-separated if G is locally finite.

Infinite gammoids
In this section, we use Theorem 1.3 to prove Theorem 1.1 and Corollary 1.2. Throughout, notation and terminology for matroids are that of [10,5]. M always denotes a matroid and E(M ) and I(M ) denote its ground set and its sets of independent sets, respectively.
Recall that the set I(M ) is required to satisfy the following independence axioms An I-circuit is a set minimal with the property that it is not in I. The following is true in any matroid.
(+) For any two finite I-circuits o 1 and o 2 and any x ∈ o 1 ∩ o 2 , there is some Given I ⊆ P(E), its finitarization I f in consists of those sets J all of whose finite subsets are in I. Usually, it is made a requirement that I is the set of independent sets of a matroid [4]. Then I f in is the set of independent sets of a finitary matroid, called M f in [4]. We shall need the following slight strengthening of this fact. Proof. Clearly I f in satisfies (I1) and (I2), and it satisfies (IM) by Zorn's Lemma. Thus it remains to check (I3). So let I, I ∈ I f in with I maximal and I not maximal. So there is some y / ∈ I with I + y ∈ I. We may assume that y / ∈ I since otherwise we are done. Thus there is some finite I-circuit o with y ∈ o ⊆ I + y. Suppose for a contradiction that for each x ∈ o\(I +y), there is some finite I-circuit o x with x ∈ o x ⊆ I +x. Applying (+) successively to o and the o x , we obtain a finite I-circuit o included in I + y, which contradicts the assumption that I + y ∈ I f in . Thus there is some x ∈ o \ (I + y) such that I + x ∈ I f in , which completes the proof.
We shall also need the following slight variation of (I3).
(*) For all I, J ∈ I and all y ∈ I \ J with J + y / ∈ I there exists x ∈ J \ I such that (J + y) − x ∈ I.  In the special case where M is finitary, N is nearly finitary. Next, we shall summarise the results from [1] that are relevant to this paper.
The digraph G AC has vertex set V AC and three types of edges: For j ∈ N it has an edge from v 1 j to b j . For each j ∈ N, it has two edges, both start at v 2 j , and end at v 1 j and v 1 j+1 . The pair (G AC , B AC ) is called an alternating comb (AC). A subdivision of AC is drawn in Figure 1. Formally, a subdivision of AC is a pair (H AC , B AC ) where H AC is obtained from G AC by replacing each directed edge xy by a directed path from x to y that is internally disjoint from all other such paths. Here edges from V 2 to V 1 are not allowed to be replaced by a trivial path 4 but the edges v 1 j b j are allowed to be replaced by a trivial path. A pair (G, B) has a subdivision of AC if there is a subgraph H AC of G and B AC ⊆ B ∩ V (H AC ) such that (H AC , B AC ) is isomorphic to a subdivision of AC. For the remainder of this section, let G denote a digraph and B ⊆ V (G). In the following, we shall explore for which digraphs G and sets B the system L T (G, B) is the set of independent sets of a matroid, and how L T (G, B) relates to L(G, B). If G is finite L(G, B) is a matroid and thus satisfies (+). The latter easily extends to infinite graphs G.  Next we prove the following slight strengthening of Corollary 1.2 from the Introduction. Below we shall refer to the definition of dominating as defined in the Introduction. Proof. Clearly, if L(G, B) is nearly finitary, there are neither infinitely many vertex-disjoint rays dominating B nor infinitely many vertices dominating B. Conversely, assume that there are neither infinitely many vertex-disjoint rays dominating B nor infinitely many vertices dominating B. L(G, B) clearly satisfies (I1) and (I2), and it satisfies (I3) and ( * ) by Lemma 5.3 and Lemma 5.4. Let J ∈ L T (G, B). By Theorem 1.3, we get for each v ∈ J a ray or path P v starting at v such that all these P v are vertex-disjoint. Moreover each such P v either ends in B or is a ray dominating B or its last vertex dominates B. Let I be the set of those v such that P v ends in B. By assumption J \ I is finite. So by Corollary 5.7, we can apply Lemma 5.2 with J = L T (G, B) to deduce that L(G, B) satisfies (IM), and thus is a nearly-finitary matroid.
A natural question that comes up is to ask how Theorem 5.5 and Corollary 5.8 relate to each other. In [1], Afzali, Law and Müller construct a pair (G, B) without AC such that L(G, B) is not nearly finitary. They also do it in a way to make L(G, B) 3-connected. Thus Corollary 5.8 does not imply Theorem 5.5.
To see that Theorem 5.5 does not imply Corollary 5.8, let G be the 3 by Z grid, formally: V (G) = {1, 2, 3}×Z, see Figure 1. In G, there is a directed edge from (x, y) to (x , y ) if and only if either x = x and y = y +1 or y = y and x = x + 1. Let: B = {3} × Z. Then it is easy to see that no vertex of G dominates B and there are not infinitely many vertex-disjoint rays dominating B. However (G, B) has a subdivision of AC, which is indicated in Figure 1. Thus, there arises the question if there is a nontrivial common generalization of Corollary 5.8 and Theorem 5.5.
During this whole section, we have only considered the directed-vertexversion. Of course, similar results are true if we consider the undirectedvertex-version, the directed-edge-version or the undirected-edge-version instead.

Acknowledgements
I am grateful to Hadi Afzali, Hiu Fai Law and Malte Müller for explaining me many things about infinite gammoids.