Krzysztof Krupiǹski pointed out to me that there is a mistake in the statement of [1, Lemma 4.9], and a gap in the proof of [1, Theorem 4.13]. In this corrigendum, the right version of the lemma and a correct proof of the theorem are presented. We use most of the definitions and notation conventions from  without explicitly formulating them.
First of all, in the statement of Lemma 4.9 there should be an additional assumption that balls , are disjoint. Then the lemma is true, its original proof goes through without any changes, and it still can be used in the proof of Theorem 4.13, which is the only place in , where Lemma 4.9 is applied.
([1, Lemma 4.9])
Let be disjoint ‘open’ balls in an ultrametric space X. Suppose that the equivalence classes they determine in , respectively, are agreeable. If , then
Suppose that , and put , , . Then there exists such that , so , and . However, , and , . Therefore cannot be agreeable. ∎
In the proof of Theorem 4.13, it is claimed that if B is an agreeable family satisfying the boundedness condition, then, for every chain L of balls coming from equivalence classes in B, either L is bounded from below in B, or it must contain a sequence of balls with diameters converging to 0. This is not true in general, so the presented construction of a d-transversal is not quite correct. The remaining part of the corrigendum is devoted to a correct proof of this theorem.
Let X be a W-space with metric d. Recall that every ball B in X gives rise to an equivalence relation on the set of all d-transversals in X, defined by
Suppose that D is an equivalence class of an equivalence relation defined as above by an ‘open’ ball B. Then D determines a unique ball in X, so we will identify such equivalence classes with the balls determined by them. In particular, by an agreeable family of balls (or a family of balls satisfying the boundedness condition), we will mean a family of balls determined by an agreeable family of equivalence classes (or a family of classes satisfying the boundedness condition.)
In order to prove the theorem, we need two auxiliary results.
Let be X a W-space, and let B be an agreeable family of balls in X which satisfies the boundedness condition. Let . Then one of the following must hold:
for every , and every ball with there is a ball such that , and ,
there is a decreasing sequence of balls in B such that , and ,
there is with which is inclusion-minimal in B.
Suppose that (i) does not hold, and fix and witnessing it. Suppose that (ii) and (iii) do not hold either. We will construct a strictly decreasing sequence of balls in B. As this is obviously not possible (because B is countable), (ii) or (iii) must hold.
Put . Suppose that we have constructed for . Suppose that . Since (iii) does not hold, is not inclusion-minimal in B, and there is with . Put .
Suppose now that α is a limit ordinal. Find a cofinal sequence in . As (ii) does not hold, the radii of are bounded from below by some . Observe that since X is locally non-rigid, r can be chosen so that, for the balls , , the relation strictly refines the relation , that is, is a covering pair in .
Fix n. Since , and , there exists such that , that is, refines . In other words, δ witnesses that covering pairs coming from are bounded from below. The boundedness condition implies that there is a ball with for every n. Put . This finishes the inductive construction. ∎
Let X be a W-space, and let B be an agreeable family of balls in X. Suppose that and are such for every ball with there is a ball such that , and . Then there is such that for every with , and for every .
Fix a maximal pairwise disjoint family of balls E such that , and . Fix for every . Because B is agreeable, Lemma 1 implies that for every . As X is a W-space, Lemma 4.10 in  implies that there exists such that for . Clearly, , and for every .
Let be a ball such that . Fix . If , then Lemma 1 implies . Otherwise , so there is such that or . In both cases , so . Suppose that there is such that . But then the ultrametric triangle inequality implies that , which is impossible. Thus, . ∎
([1, Theorem 4.13])
Let X be a W-space, and let , , for be defined as above. Then is plenary, acts transitively and faithfully on , and all elements of respect . Thus, it can be regarded as a transitive permutation group . Moreover,
Consider the action of on given by
for , .
By [1, Lemma 4.7], is a countable plenary family, and every relation in has countably many classes. By [1, Lemmas 4.10, 4.11 and 4.12], the action defined above is faithful, and is a transitive maximal permutation group respecting . Therefore, to prove the theorem, we only need to check condition (3) of Theorem 3.5 in .
Suppose that B is an agreeable family of balls in X which satisfies the boundedness condition. Let
and let .
Let be a selector for the family of all inclusion-minimal balls in B. Let be the set of all points of the form , where is a sequence of balls in B with . Then intersects every ball in , intersects every ball in , and by Lemma 1, for all .
We show how to construct intersecting every element of so that intersects every element of B, and for all .
Clearly, for every ball in , point (i) of Lemma 2 holds. Let be an enumeration of all balls in . By Lemma 3, there is such that for every with , and for every . Put . Suppose that we have constructed , so that , and is chosen as in Lemma 3 for the ball , . If for some , then put , . Otherwise, fix for as in Lemma 3, and put .
By the construction, we have that intersects every element of , and for every . By Lemma 1, is as required.
Now, by [1, Lemma 4.10], there exists a d-transversal Y such that . Obviously, A intersects every element of B. ∎
Malicki M., Separable ultrametric spaces and their isometry groups, Forum Math. 26 (2014), 1663–1683. Google Scholar
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Published Online: 2015-11-17
Published in Print: 2016-09-01