aa r X i v : . [ m a t h . GN ] N ov A CHARACTERIZATION OF ERDŐS SPACE FACTORS
DAVID S. LIPHAM
Abstract.
We prove that an almost zero-dimensional space X is an Erdős spacefactor if and only if X has a Sierpiński stratification of C-sets. We apply this char-acterization to spaces which are countable unions of C-set Erdős space factors. Weshow that the Erdős space E is unstable by giving strongly σ -complete and nowhere σ -complete examples of almost zero-dimensional F σδ -spaces which are not Erdős spacefactors. This answers a question by Dijkstra and van Mill. Introduction
All spaces under consideration are non-empty, separable and metrizable.We say that a subset A of a space X is a C-set in X if A can be written as anintersection of clopen subsets of X . A space X is almost zero-dimensional if everypoint x ∈ X has a neighborhood basis consisting of C-sets of X . A (separable metric)topology T on X witnesses the almost zero-dimensionality of X if T is coarserthan the given topology on X , ( X, T ) is zero-dimensional, and every point of X hasa neighborhood basis consisting of sets that are closed in ( X, T ). A space is almostzero-dimensional if and only if there is a topology witnessing the fact [5, Remark 2.4].Almost zero-dimensional spaces of positive dimension include the Erdős space E := { x ∈ ℓ : x n ∈ Q for all n < ω } . The almost zero-dimensionality of E is witnessed by the F σδ topology that E inheritsfrom Q ω . We call a space X an Erdős space factor if there is a space Y such that X × Y is homeomorphic to E .In 2010, Dijkstra and van Mill proved: Proposition 1 ([5, Theorem 9.2]) . For any space X the following are equivalent.(a) X is an Erdős space factor;(b) X × E is homeomorphic to E ;(c) X admits a closed embedding into E ;(d) there exists an F σδ topology witnessing the almost zero-dimensionality of X . Proposition 2 ([5, Corollary 9.3]) . Every almost zero-dimensional complete ( G δ -)spaceis an Erdős space factor. They then asked the following question, motivated by van Engelen’s result that X × Q ω ≃ Q ω for every zero-dimensional F σδ -space X [14]. Note that zero-dimensional F σδ -spaces are also Erdős space factors by condition (d) in Proposition 1. Question 1 ([5, Question 9.7]) . Is every almost zero-dimensional F σδ -space an Erdősspace factor? In this paper we give an intrinsic characterization of Erdős space factors (Theorem1) and answer Question 1 in the negative.Our characterization will imply that if X is an almost zero-dimensional space whichcan be written as a countable union of C-set Erdős space factors, then X is an Erdősspace factor (Theorem 2). For example, if X is an Erdős space factor then so is theVietoris hyperspace F ( X ) (Corollary 4). Combining Theorem 2 and Proposition 2will show that every almost zero-dimensional countable union of complete C-sets is anErdős space factor (Corollary 5). This result applies to Dijkstra’s homogeneous space T ( ˜ E, E ′ ) featured in [4] (Corollary 6).We will present three counterexamples to Question 1. One is nowhere σ -completeand F σδ , similar to E . Another is strongly σ -complete and thus automatically F σδ [15,Lemma 2.1]. All three examples are first category, yet they have the property thatcountable unions of nowhere dense C-sets have empty interiors. An Erdős space factorcannot have that combination of properties (Theorem 8).An element X of a class of topological spaces is called the stable space for that classif for every space Y in the class we have that X × Y is homeomorphic to X . The negativeanswer to Question 1 shows that E is unstable amongst the almost zero-dimensional F σδ -spaces. Likewise, the complete Erdős space E c := { x ∈ ℓ : x n ∈ P for all n < ω } is unstable in the class of almost zero-dimensional Polish spaces because E c is nothomeomorphic to its ω -power [3]. By contrast, E ω c is stable and is therefore the almostzero-dimensional analogue of P (the space of irrational numbers) [6]. It is unknownwhether a stable element exists for the class of almost zero-dimensional F σδ -spaces.2. The characterization A tree T on an alphabet A is a subset of A <ω that is closed under initial segments,i.e. if β ∈ T and α ≺ β then α ∈ T . An element λ ∈ A ω is an infinite branch of T provided λ ↾ k ∈ T for every k < ω . We let [ T ] denote the set of all infinite branchesof T . If α, β ∈ T are such that α ≺ β and dom( β ) = dom( α ) + 1, then we say that β is an immediate successor of α and succ( α ) denotes the set of immediate successorsof α in T .A system ( X α ) α ∈ T is a Sierpiński stratification of a space X if:(1) T is a non-empty tree over a countable alphabet,(2) each X α is a closed subset of X ,(3) X ∅ = X and X α = S { X β : β ∈ succ( α ) } for each α ∈ T , and(4) if λ ∈ [ T ] then the sequence X λ ↾ , X λ ↾ , . . . converges to a point in X .A space is absolute F σδ if and only if it has a Sierpiński stratification [13, Théorème]. Theorem 1.
An almost zero-dimensional space X is an Erdős space factor if and onlyif X has a Sierpiński stratification of C-sets.Proof. Suppose that X is an Erdős space factor. Then X is homeomorphic to a closedsubset of E . Since E ≃ Q ω × E c [5, Proposition 9.1], we may assume that X is aclosed subset of Q ω × E c . Let ( A α ) α ∈ S be the obvious Sierpiński stratification of Q ω in which S is a tree over Q . Let d be a complete metric for E c . For each n < ω let CHARACTERIZATION OF ERDŐS SPACE FACTORS 3 { B ni : i < ω } be a C-set covering of E c such that diam( B ni ) < /n in the metric d .Let B ∅ = E c . For each non-empty β ∈ ω <ω define B β = T { B nβ ( n ) : n < dom( β ) } .Then T := { β ∈ ω <ω : B β = ∅ } is a tree over ω , and by completeness of ( E c , d )we have that ( B β ) β ∈ T is a Sierpiński stratification of E c . If α ∈ Q <ω , β ∈ ω <ω , and n = dom( α ) = dom( β ), then we define α ∗ β = hh α (0) , β (0) i , . . . , h α ( n − , β ( n − ii . Note that S ∗ T := { α ∗ β : α ∈ S, β ∈ T, and dom( α ) = dom( β ) } is a tree over Q × ω , ( A α × B β ) α ∗ β ∈ S ∗ T is a Sierpiński stratification of Q ω × E c , andeach A α × B β is a C-set in Q ω × E c . Then (( A α × B β ) ∩ X ) α ∗ β ∈ S ∗ T is a Sierpińskistratification of X consisting of C-sets in X .Now suppose that ( A α ) α ∈ T is a Sierpiński stratification of X where every A α is aC-set in X . For each α ∈ T write A α = T { C αn : n < ω } where each C αn is clopen in X . Let { B i : i < ω } be a neighborhood basis of C-sets for X , and for each i < ω write B i = T { D ij : j < ω } where each D ij is clopen in X . The topology T that is generatedby the sub-basis { C αn , X \ C αn : α ∈ T, n < ω } ∪ { D ij , X \ D ij : i, j < ω } is easily seen to be a second countable, regular, zero-dimensional topology on X . Itwitnesses the almost zero-dimensionality of X because T is coarser than the originaltopology of X and every B i is T -closed. Further, every A α is T -closed and ( A α ) α ∈ T is aSierpiński stratification of ( X, T ). By Sierpiński’s theorem ( X, T ) is an F σδ -space. ByProposition 1, X is an Erdős space factor. (cid:3) Countable unions of C-set E -factors The power of Theorem 1 is demonstrated in the proof of the following.
Theorem 2. If X is an almost zero-dimensional space which is the union of countablymany C-set Erdős space factors, then X is an Erdős space factor.Proof. Suppose that X is almost zero-dimensional and X = S { A n : n < ω } where each A n is both an Erdős space factor and a C-set in X . By Theorem 1, for each n < ω there is a Sierpiński stratification ( B nα ) α ∈ T n of A n such that each B nα is a C-set in A n .Define a tree T = [ n<ω {h n i ⌢ α : α ∈ T n } . Put X ∅ = X and X h n i ⌢ α = B nα for each n < ω and α ∈ T n . By [5, Corollary 4.20]each B nα is a C-set in X . Thus ( X α ) α ∈ T is a Sierpiński stratification of X consisting ofC-sets in X . By Theorem 1, X is an Erdős space factor. (cid:3) By Proposition 1 and Theorem 2 we have:
Corollary 3.
Let X be an almost zero-dimensional space. If X can be written as theunion of countably many C-sets which admit closed embeddings into E , then X has aclosed embedding in E . DAVID S. LIPHAM
We will now present an application of Theorem 2. For any space X and n < ω welet F n ( X ) denote the set of all non-empty subsets of X of cardinality ≤ n . The set ofall non-empty finite subsets of X is denoted F ( X ) = [ n<ω F n ( X ) . The Vietoris topology on F ( X ) has a basis of open sets of the form h U , . . . , U k − i = { F ∈ F ( X ) : F ⊂ S i Corollary 5. If X is almost zero-dimensional space that is a countable union of com-plete C-sets, then X is an Erdős space factor. We will now apply Corollary 5 to the main example in [4]. If X is an almost zero-dimensional space and A is a C-set in X , then define T ( X, A ) = [ n<ω ( X \ A ) n × A. Let π : T ( X, A ) → X be defined by π ( x , . . . , x n ) = x . Consider the collection B ofsubsets of T ( X, A ) that consists of all sets of the form O × . . . × O n − × π − ( O n ), where n < ω , each O i is an open subset of X , and O i ⊂ X \ A for each i < n . By [4, Claim7], B forms a basis for an almost zero-dimensional topology on T ( X, A ). Corollary 6. Dijkstra’s homogeneous almost zero-dimensional space is a countableunion of complete C-sets and is therefore an Erdős space factor.Proof. The example T ( ˜ E, E ′ ) in [4] is generated by a particular complete almost zero-dimensional space ˜ E ≃ E c and a C-set E ′ in ˜ E . We claim that for X = ˜ E and A = E ′ ,the space T ( X, A ) is a countable union of complete C-sets. To prove this, for each n < ω put T n ( X, A ) = [ k ≤ n ( X \ A ) k × A. Write A as an intersection of clopen subsets of X ; A = T { C i : i < ω } . Then T n ( X, A ) = \ i<ω [ k ≤ n ( X \ A ) k × π − ( C i ) . CHARACTERIZATION OF ERDŐS SPACE FACTORS 5 For each i < ω note that S k ≤ n ( X \ A ) k × π − ( C i ) is clopen in T ( X, A ) because it is aunion of basic open sets and its complement is the basic open set ( X \ A ) n × π − ( X \ C i ).Therefore T n ( X, A ) is a C-set in T ( X, A ). Finally, each T n ( X, A ) is complete by [4,Claim 6]. (cid:3) E is unstable To prove the main result in this section, we will need the following lemma. Lemma 7. Let X be an almost zero-dimensional space, and suppose that ( A α ) α ∈ T isa Sierpiński stratification of X consisting of C-sets in X . Then there exists a Sier-piński stratification ( B β ) β ∈ S of X such that every B β is a non-empty C-set in X , and diam( B β ) < / dom( β ) .Proof. For each n < ω let { C ni : i < ω } be a C-set covering of X such that diam( C ni ) < / ( n + 1) for all i < ω . For every β = α ∗ γ ∈ T ∗ ω <ω (where the ∗ operation is definedas in the proof of Theorem 1), define B β = A α ∩ \ n< dom( α ) C nγ ( n ) . Let S = { β ∈ T ∗ ω <ω : B β = ∅ } . Then ( B β ) β ∈ S is as desired. (cid:3) Theorem 8. Every first category Erdős space factor contains a neighborhood which iscovered by countably many nowhere dense C-sets.Proof. Let X be an Erdős space factor. Suppose that no neighborhood in X can becovered by countably many nowhere dense C-sets of X . We will show that X is notfirst category. To that end, let { X i : i < ω } be any (countable) collection of closednowhere dense subsets of X . We will show X = S { X i : i < ω } .By Theorem 1 there is a Sierpiński stratification ( A α ) α ∈ T of X such that every A α is a C-set in X . By Lemma 7 we may assume that the A α ’s are non-empty anddiam( A α ) < / dom( α ) for each α ∈ T . We will now inductively define a sequence( α i ) ∈ T ω and integers N < N < . . . such that for every i < ω : (cid:4) dom( α i ) = N i , (cid:4) α i +1 ↾ N i = α i , (cid:4) A α i has non-empty interior in X , and (cid:4) A α i ∩ X i = ∅ .To begin the construction, let x ∈ X \ X . Choose N ≥ /N < ε := d ( x, X ) . Since { A α : α ∈ T, dom( α ) = N , and A α ∩ B ( x, ε/ = ∅ } is a countable C-setcovering of the neighborhood B ( x, ε/ α ∈ T such that dom( α ) = N , A α contains a non-empty open subset of X , and A α ∩ B ( x, ε/ = ∅ . The lastcondition implies A α ∩ X = ∅ because if y ∈ A α then d ( x, y ) ≤ ε/ A α ) < ε/ / dom( α ) = ε/ /N < ε. Now suppose α i ∈ T and N i have been appropriately defined for a given i < ω . Since A α i contains a non-empty open subset of X , and X i +1 is nowhere dense in X , there DAVID S. LIPHAM exists x ∈ X and ε > B ( x, ε ) ⊂ A α i \ X i +1 . Choose N i +1 > N i such that2 /N i +1 < ε . Observe that { A α : α ∈ T, dom( α ) = N i +1 , α ↾ N i = α i , and A α ∩ B ( x, ε/ = ∅ } covers B ( x, ε/ α i +1 ∈ T such that dom( α i +1 ) = N i +1 , α i +1 ↾ N i = α i , A α i +1 has non-empty interior in X , and A α i +1 ∩ B ( x, ε/ = ∅ (hence A α i +1 ∩ X i +1 = ∅ ).Thus the construction can be continued.Finally, let λ = [ i<ω α i ∈ [ T ] . By the convergence property (4) in the definition of a Sierpiński stratification, the sets A λ ↾ , A λ ↾ , . . . converge to a point x λ ∈ X . Then x λ ∈ \ n<ω A λ ↾ n = \ i<ω A α i ⊂ X \ [ i<ω X i . Therefore X = S { X i : i < ω } . (cid:3) We are now ready to present counterexamples to Question 1.Our first example is elementary and will require two basic lemmas concerning thetopologies of ℓ and E c . Lemma 9. Let x, x , x , . . . ∈ ℓ . Suppose x n → x in R ω and k x n k < k x k + 1 /n foreach n < ω . Then x n → x in ℓ .Proof. By coordinate-wise convergence of ( x n ) and the lemma in [12], it suffices to show k x n k → k x k . To that end, let ε > 0. Since { y ∈ R ω : k y k > k x k − ε } contains x and isopen in R ω , there exists N such that 1 /N < ε and k x n k > k x k − ε for all n ≥ N . Then k x k − ε < k x n k < k x k + ε for all n ≥ N . Thus k x n k → k x k . (cid:3) Lemma 10. E c has a basis of neighborhoods of the form { x ∈ C : k x k ≤ q } where C is clopen in P ω and q ∈ Q .Proof. Let y ∈ E c and let U be any open subset of E c with y ∈ U . There exists( q n ) ∈ Q ω such that k y k < q n < k y k + 1 /n for every n < ω . There is also a sequence C ⊃ C ⊃ . . . of clopen subsets of P ω which converges to y in the topology of P ω . If x n ∈ { x ∈ C n : k x k ≤ q n } for every n < ω , then x n → y by Lemma 9. Thus thereexists n < ω such that { x ∈ C n : k x k ≤ q n } ⊂ U . Clearly { x ∈ C n : k x k ≤ q n } is an E c -neighborhood of y . (cid:3) Let E ′ = { x ∈ ℓ : x n ∈ Q + √ n < ω } . Example 1. X := E ′ ∪ { x ∈ E c : k x k ∈ Q } is an almost zero-dimensional F σδ -spacewhich is not an Erdős space factor.Proof. Note that E ′ ≃ E and { x ∈ E c : k x k = q } is nowhere dense in X for each q ∈ Q .So X is first category. Further, X is the union of two F σδ -subsets of E c and is thereforealmost zero-dimensional and absolute F σδ . In order to apply Theorem 8, we need toshow that no neighborhood in X can be covered by countably many nowhere denseC-sets of X . Let A be any neighborhood in X . By Lemma 10 we may assume that A = { x ∈ C : k x k ≤ q } where C is clopen in P ω and q ∈ Q . For a contradiction, CHARACTERIZATION OF ERDŐS SPACE FACTORS 7 suppose that A ⊂ S { A n : n < ω } where each A n is a nowhere dense C-set in X . Notethat { x ∈ C : k x k = q } is complete, and its topology as a subspace of X is the same asthe topology it inherits from P ω . By Baire’s theorem there is a clopen set B ⊂ C and n < ω such that ∅ = { x ∈ B : k x k = q } ⊂ A n . Then the open set { x ∈ B : k x k < q } is also non-empty. Since A n is nowhere dense, there exists x ∈ X ∩ B \ A n such that k x k < q . Let O be a clopen subset of X such that x ∈ O and O ∩ A n = ∅ . Then O ∩ A ∩ E ′ is a non-empty bounded clopen subset of E ′ . This contradicts a key propertyof the Erdős space [7], namely that E ′ ∪ {∞} is connected. (cid:3) Our next two examples are from complex dynamics. Define f ( z ) = exp( z ) − z ∈ C . The Julia set J ( f ) is a Cantor bouquet of rays in the complex plane [1],and has a natural endpoint set E ( f ) which is almost zero-dimensional and complete.In fact, E ( f ) is homeomorphic to E c [8]. We let f n = f ◦ f ◦ . . . ◦ f | {z } n times denote the n -fold composition of f .Recall that a space X is nowhere σ -complete if no neighborhood in X can bewritten as a countable union of complete subspaces. This is equivalent to saying thatno neighborhood in X is absolute G δσ (i.e. X is nowhere G δσ ). Example 2. There is an almost zero-dimensional F σδ -space that is nowhere G δσ andis not an Erdős space factor.Proof. The escaping endpoint set ˙ E ( f ) := { z ∈ E ( f ) : f n ( z ) → ∞} is almost zero-dimensional and first category [9], F σδ and nowhere G δσ [11], and no neighborhood in˙ E ( f ) can be covered by countably many nowhere dense C-sets of ˙ E ( f ) [10, Remark5.3]. By Theorem 8, ˙ E ( f ) is not an Erdős space factor. (cid:3) By contrast, a space X is strongly σ -complete if X can be written as a countableunion of closed complete subspaces. Example 3. There is a strongly σ -complete almost zero-dimensional space which isnot an Erdős space factor.Proof. Consider the set ˆ E ( f ) := { z ∈ E ( f ) : { f n ( z ) : n < ω } 6 = J ( f ) } consisting ofall endpoints whose (forward) orbits are not dense in J ( f ). By [2, Lemma 1], ˆ E ( f )is a first category F σ -subset of E ( f ). In particular, ˆ E ( f ) is strongly σ -complete andabsolute F σδ . Clearly ˙ E ( f ) ⊂ ˆ E ( f ), and ˙ E ( f ) is dense in E ( f ) by Montel’s theorem.Thus no neighborhood in ˆ E ( f ) can be covered by countably many nowhere dense C-setsof ˆ E ( f ). By Theorem 8, ˆ E ( f ) is not an Erdős space factor. (cid:3) Remark 1. In light of Corollary 5, ˆ E ( f ) is an example of a strongly σ -complete almostzero-dimensional space which cannot be written as a countable union of complete C-sets. Remark 2. van Mill proved that Q × P is the unique zero-dimensional space whichis strongly σ -complete, nowhere σ -compact, and nowhere complete [16]. There is nosuch classification of almost zero-dimensional spaces, as ˆ E ( f ) is strongly σ -complete,nowhere σ -compact, nowhere complete, and is not a Q -product. DAVID S. LIPHAM References [1] J. M. Aarts, L. G. Oversteegen, The geometry of Julia sets, Trans. Amer. Math. Soc. 338 (1993),no. 2, 897–918.[2] I. N. Baker and P. Domínguez, Residual Julia Sets, Journal of Analysis, Vol. 8, 2000, pp. 121–137.[3] J. J. Dijkstra, J. van Mill and J. Steprans, Complete Erdős space is unstable, MathematicalProceedings of the Cambridge Philosophical Society 137 (2004), 465–473.[4] J. J. Dijkstra, A homogeneous space that is one-dimensional but not cohesive, Houston J. Math.32 (2006), no. 4, 1093–1099.[5] J. J. Dijkstra, J. van Mill, Erdős space and homeomorphism groups of manifolds, Mem. Amer.Math. Soc. 208 (2010), no. 979.[6] J.J. Dijkstra, Characterizing stable complete Erdős space, Israel J. Math. 186 (2011) 477–507.[7] P. Erdős, The dimension of the rational points in Hilbert space, Ann. of Math. (2) 41 (1940),734–736.[8] K. Kawamura, L. G. Oversteegen, and E. D. Tymchatyn, On homogeneous totally disconnected1-dimensional spaces, Fund. Math. 150 (1996), 97–112.[9] D. S. Lipham, A note on the topology of escaping endpoints, Ergodic Theory Dynam. Systems,(2020) https://doi.org/10.1017/etds.2019.111 .[10] D. S. Lipham, Distinguishing endpoint sets from Erdős space, arXiv preprint (2020); https://arxiv.org/pdf/2006.04783.pdf .[11] D. S. Lipham, Another almost zero-dimensional space of exact multiplicative class 3, arXivpreprint (2020); https://arxiv.org/pdf/2010.13876.pdf .[12] J. H. Roberts, The rational points in Hilbert space, Duke Math. J. 23 (1956), 488–491.[13] W. Sierpiński, Sur une définition topologique des ensembles F σδ , Fund. Math. 6 (1924), 24–29.[14] F. van Engelen, Countable products of zero-dimensional absolute F σδ spaces, Indag. Math. 87:4(1984) 391–399.[15] F. van Engelen and J. van Mill, Borel sets in compact spaces: Some Hurewicz-type theorems,Fund. Math. 124 (1984), 271–286.[16] J. van Mill, Characterization of some zero-dimensional separable metric spaces, Trans. Amer.Math. Soc. 264 (1981), 205–215.[17] A. Zaragoza, Symmetric products of Erdős space and complete Erdős space, Topology Appl. 284(2020), 1–10. Department of Mathematics, Auburn University at Montgomery, Montgomery AL 36117,United States of America Email address ::