aa r X i v : . [ m a t h . GN ] J u l On zero dimensional sequential spaces
Paul Fabel
Department of Mathematics & StatisticsMississippi State, MS,[email protected]
Abstract
We develop tools to recognize sequential spaces with large inductivedimension zero. We show the Hawaiian earring group G is 0 dimensional,when endowed with the quotient topology, inherited from the space ofbased loops with the compact open topology. In particular G is T andhence inclusion G ֒ → F M ( G ) is a topological embedding into the freetopological group F M ( G ) in the sense of Markov. Acknowledgements
The author gratefully acknowledges partial support fromUniversity Primorska, Feb 22-August 1 2020
When is a Hausdorff sequential space zero dimensional? The fundamentalgroup of the Hawaiian earring serves as catalyst for such an inquiry in thecontext of the following three questions. We answer the first questionaffirmatively via partial answers to second and third.1) Is the large inductive dimension of the Hawiian earring group zero,if G enjoys the quotient topology inherited from the space of based loops?2) If a sequential space G continuously injects into a countable inverseof limit of discrete spaces, what conditions ensure G is zero dimensional?3) If a sequential space G is a quotient of a countable product ofdiscrete spaces, what conditions ensure G is zero dimensional?Wild algebraic topology is loosely described as the study of locallycomplicated spaces, and their attendant homotopy/homology groups. Themotive to impose a topology on the latter objects might come from func-torality of the fundamental group [6], from a canonical bijection between π ( X, p ) and the fibres of a semicovering E → X [23][26][27], or to measurethe extent to which π might act continuously on a space [8].At center stage [17][10][18][14][11][13] is the Hawaiian earring HE , anull sequence of loops joined at a common point, the inverse limit ofnested sequence of bouquets on n loops, under retraction bonding maps,collapsing the nth loop to the special point p. The induced homomorphism φ : π ( HE, p ) → lim ← F n , with F n thediscrete free group on n generators, is one to one [15][24][22]. Thus the ubgroup im ( φ ) determines a sense in which elements of π ( HE, p ) canbe understood as precisely the “infinite irreducible words” in the letters x , x − , x . . . . so that each letter appears finitely many times [10].To impose a topology on π ( HE, p ), at one extreme we might insistthat φ is a topological embedding. This creates a zero dimensional topo-logical group, but with the drawback that the topology is permissive forwhat is allowed to converge. For example the sequence ( x x n x − x n ) n → ∈ π ( HE, p ) , but all corresponding lifts diverge, with the topology ofuniform convergence.At another extreme, invoking a construction similar to the familiaruniversal cover, [4][22], the group π ( HE, p ) acts freely and isometricallyon a corresponding generalized universal cover of HE , a uniquely arcwiseconnected, locally path connected metric space, i.e. a topological R-tree R . The trade off here is that it is difficult for our isometries to con-verge. Treated as a group of isometries of R , π ( HE, p ) fails to be even aquasitopological group, we can have x n → id with { x n x } diverging.One compromise is to impose the quotient topology on π ( HE, p ) , defined as a quotient of the space of based loops in HE with the uniformtopology. This resolves two of the mentioned drawbacks, the sequence { ( x x n x − x n ) n } now diverges, and group translation is now continuous.On the other hand this comes at the cost of metrizability. The subspace { , ( x x n x − x n ) m } ⊂ π ( HE, p ) is a Frechet Urysohn fan [1] [20] andhence π ( HE, p ) is not first countable. This begs the question of whichfamiliar separation axioms does π ( HE, p ) satisfy?Continuity of the injection φ : π ( HE, p ) → lim ← F n ensures π ( HE, p )is T , since the codomain is T . Morever π ( HE, p ) is a quotient of aseparable metric space and hence π ( HE, p ) is Lindelof. Unfortunately φ is not a topologial embedding [19], and π ( HE, p ) is not a topologicalgroup in TOP [20]. This calls into question whether π ( HE, p ) is at least T , and hence T , since π ( HE, p ) is a Lindelof space. To prove π ( HE, p )is T it suffices to prove π ( HE, p ) has large inductive dimensions zero,that disjoint closed sets can be thickened into disjoint clopens.While the class of contractible space shows dimension is generally notan invariant of the homotopy type of an underlying space, functoralityensures π ( X ) and π ( Y ) have the same dimension if X and Y are ho-motopy equivalent. More esoterically, the knowledge that π ( HE, p ) iszero dimensional will ensure for example, that π ( HE, p ) cannot containa copy of the totally disconnected 1 dimensional Erdos space [16].To prove π ( HE, p ) is zero dimensional we establish Theorems 2.15and 3.1, applicable to suitably well behaved quotients of the inverse limitof countably many discrete nested retracts.The potential difficulty of such an inquiry is highlighted by the familiardimension raising closed quotient of the Cantor set { , } × { , } . . . → [0 ,
1] mapping each binary sequence onto the corresponding real number.What goes wrong? All finite approximations to (0 , , , . ) and (1 , , . . . )are distinct, yet the points are identified in the limit. This is analogous toa failure of π injectivity in shape theory. To avoid this, in this paper weonly consider quotients where the above phenomenon does not happen,and in particular the above does not happen in the Hawaiian earring.Most of the paper is devoted to a proof of the following. Suppose ⊂ X . . . is a nested sequence of discrete retracts X n +1 → X n and q n : X n → G n is a quotient map. This data induces a quotient map q : lim ← X n → G , and the following two extra assumptions ultimatelyensure that q does not raise dimension. 1) The retractions X n +1 → X n induce a map G n +1 → G n and 2) The inclusion maps X n → X n +1 inducea map G n → G n +1 . Our main result (Theorems 2.15 and 3.1) is that aspace G constructed in this manner has large inductive dimension zero.The proof uses the well ordering principle, and we now indicate why wellordering is useful to circumvent the failure of a less sophisticated approach.The proof idea for Theorems 2.15 and 3.1 stems from an easier butstill complicated construction designed to prove that the Hawaiian earringgroup π ( HE, p ) is a T space. To prove that π ( HE, p ) is a T spaceit suffices to prove that π ( HE, p ) has a basis of clopen sets, i.e. that π ( HE, p ) has small inductive dimension 0. In turn, since π ( HE, p ) ishomogeneous, it suffices, to start with a closed set B ⊂ π ( HE, p ) sothat the identity e / ∈ B, and thicken B into a clopen set U ( B ) so that B ⊂ U ( B ) and e / ∈ U ( B ). We now outline the overall strategy for building U ( B ) and point out a potenially fatal pitfall.By construction π ( HE, p ) = G is equipped with a canonical countablecollection of clopen sets U ( g n ) ⊂ π ( HE, p ), the preimage of g n ∈ G n under the retraction G → G n , here G n is the free group on n generators.The overall idea it to somehow thicken B ⊂ π ( HE, p ) into a union U ( B )of our special clopens, so that U is clopen and e / ∈ B .Given a closed set B ⊂ π ( HE, p ), the simplest idea to construct U ( B ) is to exploit the God given retraction R : G \{ e } → ( S G n ) \{ e } .Unfortunately this does not work, as indicated below, suggesting whymore refined methods for building U ( B ) are needed.Consider the retraction R : G \{ e } → ( S G n ) \{ e } taking the nontrivialword g ∈ G to R ( g ) = g n (deleting all letters greater than n from g ) with n minimal so that g n is nontrivial. Given B ⊂ G closed with e / ∈ B it is at least plausible that S g ∈ B U ( R ( g )) is clopen, but the followingexample shows this is false, e can be a missing limit point. Supposefor k > B is the sequence of finite words { w ( k ) } with w ( k ) = ( x x k +1 x − x k +1 ) k x k . Thus R ( w ( k )) = x k and hence e is a missinglimit point of the union of the sets U ( x k ) . For the latter example the reader might notice that we could get an ac-ceptable thickening U ( B ) using the union of the sets U ( x x k +1 x − x k +1 ) k x k ) , i.e. given b ∈ B we should look for a large index approximation of b, ratherthan a small index approximation of b. This is indeed the right idea, butcomes at the cost of no obvious best method to approximate b , we havetoo many choices as shown by the example b = x x x . . . . ∈ π ( HE, p ).In other words the best way to approximate b ∈ B is context dependent,depending both on B and also previously made choices when attemptingto thicken some of B .To make the previous sentence more precise, we indicate a more sys-tematic way to thicken B ⊂ π ( HE, p ) into a clopen. The overarch-ing idea is to impose a linear order on π ( HE, p ) (not compatible withits topology, and not well ordered), but so that nevertheless each closedset A ⊂ π ( HE, p ) has a minimal element. Given such an ordering, westart with the minimal b ∈ B, then thicken b into U ( R ( b )), then let the minimal element of the closed set B \ U ( R ( b )) . Then we select U ( R ( b )) and so on. Crucially we hope to ensure the union of the se-quence U ( R ( b )) S U ( R ( b )) S . . . is clopen.If we are suitably careful with the definition of our linear order on G ,Lemma 2.11 ensures that, proceeding by transfinite recursion, the unionof our selected sets U ( R ( b i )) is indeed the desired clopen set U ( B ) . Thisidea is the key to the paper, and we hope it will remain undisguised bythe superficially technical appearance of its implementation.Corollary 3.2 shows π ( HE, p ) is indeed 0 dimensional and in partic-ular π ( HE, p ) is a Tychonoff space (completely regular). This pays offcategorically both in TOPGRP [2] and the (compactly generated groups) k − GRP [25].On the one hand we might ingressively blame the failure of π ( HE, p )to be a topological group in TOP on an abundance of closed sets (The-orem 1 [3]). As summarized in section 3 [6], can π ( HE, p ) be repairedcategorically, by coarsening π ( HE, p )to be the canonical quotient of thefree Markov topological group F M ( π ( HE, p )). The payoff here is thatour new knowledge that π ( HE, p ) is a Tychonoff space contributes toour understanding of the latter construction. We are assured that inclu-sion π ( HE, p ) → F M ( π ( HE, p )) is a topological embedding, as noted inLemma 3.1 [6]. See also [2].More congressively [12], instead of coarsening π ( HE, p ) , we mightinstead accept π ( HE, p ) as a 0-dimensional first class citizen in k − SEQGROUP , the category of sequential spaces with group structure,so that the group operations are sequentially continuous, and acknowl-edge that the familiar product topology is categorically not always themost useful way to multiply spaces [9] [5].Looking ahead the hope is that the tools developed in this paper willprove useful to answer more general questions such as “If X is a planar con-tinuum, with the quotient topology must π ( X, p ) be zero dimensional?”We conjecture the answer is “yes”. G is zero dimen-sional The purpose of this section is to prove Theorem 2.15, every space G satisfyingthe three axioms below has large inductive dimension zero.We do not assume G is the Hawaiian earring group, but the readermay find it helpful to assume otherwise, to assume G n is the free groupof maximally reduced words on n generators, and to treat a nontrivialelement g ∈ G as an “irreducible” countably infinite word in the letters x , x − , x , . . . so that each letter appears finitely many times, and so thateach subinterval of letters in g represents a nontrivial loop in the Hawaiianearring.Constructing a useful linear order on G is a complicated affair carriedout in detail in section 3, but the rough idea is the following. Assuming G is the Hawaiian earring group, to compare g ∈ G and h ∈ G , firstdelete all letters except x and x − , but don’t reduce. If the surviving nreduced words are different this is adequate to tell g and h apart. Wecan also arrange that homotopy classes of words in x , x − G . Having well ordered the unreduced wordsin x , x − , we then extend the well ordering to the unreduced words in { x , x − , x , x − } so as to ensure axioms 2.2 and 2.3 are destined to hold.The important point is that to compare g and h we look, successively attheir unreduced approximations, until we find the minimal index wherethey differ as unreduced words. In particular G does not have the lexi-graphic order determined by G n . axiom 2.1. We assume G is a sequential space (a space so that if A ⊂ G is not closed, there exsts a convergent sequence a n → x so that a n ∈ A and x / ∈ A ). We assume G ⊂ G .... is a nested sequence of closeddiscrete subspaces and for each n ∈ { , , .. } the map Π n : G → G n is aretraction. We assume the canonical map φ : G → Π G n is one to one,defined as φ ( g ) = { Π n ( g ) } . We we do NOT require that φ is a topologicalembedding and we do NOT require that φ is a surjection. We assume Π n − = Π n − Π n and we assume the sequence Π n ( g ) → g pointwise foreach g ∈ G . axiom 2.2. We assume G admits a linear order < so that every closedset B ⊂ G has a minimal element, so that Π ( g ) ≤ Π ( g ) .... ≤ g , so thateach subspace G n is well ordered, and for each strictly increasing sequence g < g .... in G , either every subsequence of { g n } diverges, or lim g n =sup { g n } . (We do NOT require that G, < has the order topology, and wedo NOT assume that the discrete subspace G n has the order topology). axiom 2.3. Define G ∞ = G ∪ G , . . . and given k ∈ G ∞ obtain N minimal so that k ∈ G N and define Blowup ( k ) = Π − N Π N ( k ) . We assumeif k < k < k with each k n ∈ G ∞ then if Blowup ( k ) ⊂ Blowup ( k ) then Blowup ( k ) ⊂ Blowup ( k ) . Remark . Since each space G n is discrete the countable product Π ∞ n =1 G n is T . Hence, G is also T since φ : G ∞ → lim ← G n is continuous and oneto one (altough typically NOT a topological embedding). In particularconvergent sequences in G have unique limits. Lemma 2.5.
The restriction φ | G ∞ maps G ∞ bijectively onto the eventu-ally constant sequences in lim ← G n . (In general φ | G ∞ is NOT a topologiclembedding.)Proof. Given g ∈ G ∞ obtain M minimal so that g ∈ G M . If M ≤ M + n then G M ⊂ G M + n , and since Π M + n is a retraction we have Π M + n ( g ) = g .Thus φ ( g ) is eventually constant.Conversely suppose g ∈ G and φ ( g ) is eventually constant. Obtain N minimal so that Π N + n ( g ) = Π N ( g ) if N ≤ N + n . By axiom2.1 Π n ( g ) → g and by Remark2.4 Π N ( g ) = g. Thus sinceΠ N is a retraction g ∈ G N and hence g ∈ G ∞ . emma 2.6. Suppose { a, b } ⊂ G and a < b . Then there exists N so thatif N ≤ n , then a < Π n ( b ) .Proof. By axioms 2.1 we know Π n ( b ) → b and also Π ( b ) ≤ Π ( b ).... ≤ b .Thus if the result were false, and since < is a linear order, we would haveΠ n ( b ) ≤ a for all n . Hence since sup b n = b by axiom 2.2, we would obtainthe contradiction b ≤ a . Lemma 2.7.
Suppose V ⊂ G is clopen and V < b . Then there exists N so that V < Π n ( b ) if N ≤ n .Proof. Since the sequence Π n { b } is nondecreasing (axiom 2.2) it sufficesto find N so that V < Π N ( b ). To get a contradiction suppose no such N exists. For each n obtain k n ∈ V so that Π n ( b ) ≤ k n . If there exists k N sothat Π n ( b ) ≤ k N for all n , then b ≤ k N by axiom 2.2. But since k N ∈ V ,this would contradict the assumption that V < b . Thus no such k N existsand hence for each N the inequality Π n ( b ) ≤ k N as only finitely manysolutions. Hence, starting at k , we can recursively manufacture inter-leaved subsequences k < b n ≤ k n < b n ≤ k n ..... Thus by axiom 2.2both subsequences converge to the same limit, and in particular { k n } hasa subsequence converging to b . Since V is closed, we get the contradiction b ∈ V . Lemma 2.8. If U ⊂ G is nonempty and clopen in G , then minU ∈ G ∞ .If the convergent strictly increasing sequence g < g ... → g then g / ∈ G ∞ and in particular there exists N so that if N ≤ n < m then Π n ( g ) < Π m ( g ) < g .Proof. Suppose U ⊂ G is nonempty and clopen. By axiom 2.2 minU exists. Suppose b ∈ U \ G ∞ . Since U is open and since Π n ( b ) → b (axiom2.1), obtain N so that Π N ( b ) ∈ U . By axiom 2.2 Π n ( b ) ≤ b and sinceΠ N ( b ) ∈ G N ⊂ G ∞ and since b / ∈ G ∞ we have Π N ( b ) < b . Hence b is notminimal in U .If g ∈ G N , then since Π N ≤ id | G N , g is minimal in the clopen set V =Π − N Π N ( g ). Since V is open and since g is minimal in V it is impossiblethat there exists a convergent sequence g < g .... → g . Definition 2.9. If L and H are linealry ordered sets a function f : L → H is strictly increasing if, given s < t in L , if x ∈ f ( s ) and y ∈ f ( t ) ,then x < y . Lemma 2.10.
Let S ⊂ G denote the collection of clopen sets in G andsuppose [0 , i ) is an intital segment of the well ordered set J . Suppose γ : [0 , i ) → S is strictly increasing and suppose V ( j ) = ∪ k ≤ j γ ( k ) is clopenfor each j < i . Then V ( i ) = ∪ k ≤ i γ ( k ) is missing at most one limit point x . If so, there exists an increasing s < s ... sequence terminal in [0 , i ) .Moreover for any terminal increasing sequence t < t < ... in [0 , i ) , if x n ∈ γ ( t n ) then x n → x with x = sup ( V ( i )) . roof. Note V ( i ) is open in G . If V ( i ) is not closed in G , then, since G is a sequential space (axiom 2.1) suppose { k n } ⊂ V ( i ) is a convergentsequence so that k n → x / ∈ V ( i ). Suppose a n is also a convergent sequencein V i with a n → y / ∈ V ( i ). To prove V ( i ) is missing precisely one limitpoint it suffices to show that x = y , since this will ensure V ( i ) ∪ { x } issequentially closed in the sequential space T space G (axiom 2.1), andhence that V ( i ) ∪ { x } is closed in G .Note { k n } admits no constant subsequence since otherwise we get thecontradiction x ∈ γ ( s ) for some s . Thus we may refine so that the terms of { k n } and { a n } are distinct. Moreover since [0 , i ) is well ordered, and since γ i is increasing, we may further refine so that both sequences are strictlyincreasing. Thus wolog k n ∈ γ i ( s n ) and a n ∈ γ i ( t n ) with s < s ... and t < t ...,.Note { s n } is unbounded in [0 , i ) since otherwise we get the followingcontradiction. Let sup { s n } = s < i . Then V ( s ) is a closed subspaceof G and hence x ∈ V ( s ) ⊂ V ( i ). Thus both sequences { s n } and { t n } are unbounded in the well ordered set [0 , i ) and hence the sequences areinterlaced. It follows from axiom 2.2 that x = y and x = sup ( V ( i )). Lemma 2.11.
Suppose [0 , i ) is an initial segment of the well orderedset J , suppose S denotes the clopen subsets of G . Suppose the functions κ : [0 , i ) → G and γ : [0 , i ) → S are strictly increasing. Suppose is afunction K : [0 , i ) → S. Suppose V ( j ) = ∪ k ≤ j γ ( k ) is clopen in G for each j < i . Suppose for each j < i we have γ ( j ) = (Π − η ( j ) Π η ( j ) ( κ ( i ))) \ K ( j ) with K ( j ) clopen in G and κ ( j ) < K ( j ) . Suppose given j, κ ( j ) and K ( j ) , theindex η ( j ) is minimal to ensure that γ | [0 , j ] is increasing. Suppose K ( j ) is eventually constant, i.e. there is a clopen set K ⊂ G and λ ∈ [0 , i ) sothat K ( j ) = K if λ ≤ j < i . Then V ( i ) = ∪ j
Suppose a ∈ G and a < B with B ⊂ G a nonemptyclosed set. The following proof constructs a clopen set V ( a, B ) ⊂ G sothat a < V ( a, B ) and B ⊂ V ( a, B ) . Moreover if the convergent increasingsequence a < a ... → a there exists M so that if M ≤ n then V ( a m , B ) = V ( a, B ) .Proof. Let J be a well ordered set with minimal element 0 so that G ∞ < | J | . By axiom 2.2 let b = minB . Apply Lemma 2.6 and obtain N minimal so that a < Π N ( b ). Define γ (0) = Π − N (Π N ( b )). We will useLemma 2.11 repeatedly, in the special case K j = ∅ for all j .Let S denote the clopen subsets of G . Suppose i ∈ J and γ | [0 , i ) → S is strictly increasing, suppose for each j ≤ i the set V ( j ) = ∪ k ≤ j γ ( k )is clopen in G . Suppose for each j < i we have γ ( j ) = Π − n j Π n j ( c j ).Suppose given j and c j the index n j is minimal to ensure that γ | [0 , j ] isincreasing. Thus by Lemma 2.11 the set W ( i ) = ∪ j
Suppose X is a space and the set [0 , i ] is well orderedwith the order topology and suppose there exists a terminal sequence s
2. Given κ ( i ) ∈ U , get the mentioned j U , and note withfinitely many exceptions we have j U < t n . 2 ⇒ U ⊂ X with κ ( i ) ∈ U but so that V = f − ( U ) contains no open right ray. Starting with n = 1 and proceedingrecursively, for each n obtain t n > s n and t n > t n − so that t n / ∈ V . Thus t < t . and { t n } is terminal in [0 , i ). Thus with finitely many exceptions f ( t n ) ∈ U and hence t n ∈ V eventually. This constradicts t n / ∈ V . G has large inductive dimension 0 Theorem 2.15. G has large inductive dimension .Proof. Let J be a well ordered set with minimal element 0 so that | J |≥| G ∞ | . Suppose A and B are disjoint nonempty closed sets in G . De-fine κ (0) = min ( A ∪ B ). If κ (0) ∈ A define B (0) = B , and define K (0) = V ( κ (0) , B (0)), the clopen set from Theorem 2.13. If κ (0) ∈ B define A (0) = A and, again using the construction from Theorem 2.13,define K (0) = V ( κ (0) , A ). Define γ (0) = Π − Π ( κ (0)) \ K (0).Transfinite induction hypothesis: Suppose i ∈ J and γ : [0 , i ) → S satisfies, with one possible exception, all of the hypotheses of Lemma 2.11,but not necessarily the requirement that K ( j ) is eventually constant.To be precise suppose κ : [0 , i ) → A ∪ B is a function. Suppose V ( j ) = ∪ k ≤ j γ ( k ) is clopen in G for each j < i . Suppose for each j < i we have γ ( j ) = (Π − η ( j ) Π η ( j ) ( κ ( j ))) \ K ( j ) with K ( j ) clopen in G and ( j ) < K ( j ). Suppose given j, κ ( j ) and K ( j ) the index η ( j ) is minimalto ensure that γ | [0 , j ] is increasing. Let U ( j ) = (Π − η ( j ) Π η ( j ) ( κ ( j ))). Wealso assume for all j < i that γ ( j ) ∩ A = ∅ or γ ( j ) ∩ B = ∅ . If j < iW ( j ) = ∪ k Theorem 3.1. Suppose for each n ∈ { , , , . . . . } , X n is a discrete space.We assume X ⊂ X . . . . and for each n the map R n : X n +1 → X n is aretraction. Let the space X ∞ = lim ← X n denote the topological inverselimit, i.e. X ∞ is the subspace of the countable product X × X × . . . sothat ( x , x , . . . ) ∈ X ∞ iff R n ( x n +1 ) = x n for each n. Suppose furthermore we have a sequence of quotient maps q n : X n → G n so that the formula q n R n q − n +1 = r n induces a map such that r n q n +1 = q n R n , i.e. there is an induced map r n : G n +1 → G n , commuting withthe retraction X n +1 → X n . Finally suppose the formula q n +1 q − n inducesan embedding j n : G n → G n +1 , commuting with inclusion X n → X n +1 . By definition the maps { q n } induce an equivalence relation on X ∞ such that ( x , x , . . . ) ∼ ( y , y , . . . ) iff q n ( x n ) = q n ( y n ) for each n. Define G as the corresponding topological quotient q : X ∞ → G . Then G haslarge inductive dimension 0.Proof. Our strategy is to apply Theorem 2.15 by first showing G can bemade to satisfy axioms 2.1, 2.2 and 2.3. For axioms 2.2 and 2.3, we mustbuild a linear order on X ∞ which induces a suitable linear order on G .This is ultimately straightforward, but with a few restrictions imposedby the starting data { q n } , the need to ensure axiom 2.3, and the need to nsure that R n ≤ id | X n +1 . To define a lexical order on X ∞ it suffices toproceed recursively, first defining a well ordering of X , then extending toa well ordering of X and so on.To impose a well ordering on X , first arbitrarily well order G , andthen arbitrarily well order each point primage under q : X → G .Now we define on X a kind of local lexical order as follows. Tocompare two points { x , y } ⊂ X , if q ( x ) = q ( y ) let the order in G determine which is bigger. If q ( x ) = q ( y ) let the order on pointpreimages of q decide which is bigger. Crucially if q ( y ) = q ( x ) and q ( x ) = q ( z ) then y < { x , z } or y > { x , z } .To extend the well ordering of X to X we begin as follows. First, foreach x ∈ X define X ( x ) = X ∩ R − ( x ) , and note x ∈ X ( x ) since X ⊂ X and R ( x ) = x . Next, define G ( x ) = q ( X ( x )) and note G ( x ) ⊂ G . Now well order G ( x ) to have minimal element q ( x ) andotherwise the well ordering of G ( x ) is arbitrary. Next, well order eachpoint preimage of the map q | X ( x ) : X ( x ) → G ( x ) subject only tothe constraint that x = min q ( x ) . Thus x = min { X ( x ) } .To complete the definition of the well ordering on X suppose weare,given distinct points { x , y } ⊂ X . If R ( x ) = R ( y ) we require theorder of X to dictate which is bigger. If R ( x ) = R ( y ) and q ( x ) = q ( y ) we require the order of G ( R ( x )) to dictate which is bigger. If R ( x ) = R ( y ) and q ( x ) = q ( y ) we require the order on q ( x ) todictate which is bigger. In summary, lexical inspection of the orderedtriples ( R ( x ) , q ( x ) , x ) and ( R ( y ) , q ( y ) , y ) determines which of { x , y } is bigger.Crucially, if { x , y , z } ⊂ X and x < y < z and q R ( x ) = q R ( z ) , then q R ( x ) = q R ( y ), and we argue the contrapositive asfollows. Suppose q R ( x ) = q R ( y ) and q R ( x ) = q R ( z ) with { x , y , z } ⊂ X . Let x = q ( x ) and y = q ( y ) and z = q ( z ). Asnoted we must have y < { x , z } or y > { x , z } , and hence by definition y < { x , z } or y > { x , z } . Finally note R ≤ id | X . Proceeding recursively, suppose X n − has been well ordered so thatif q n − ( x n − ) = q n − ( y n − ) and q n − ( x n − ) = q n − ( z n − ) theny n − < { x n − , z n − } , or y n − > { x n − , z n − } . Suppose R n − ≤ id | X n − . For each x n − ∈ X n − define X n ( x n − ) = X n ∩ R − n − ( x n − ), andnote x n − ∈ X n ( x n − ) since X n − ⊂ X n and R n − ( x n − ) = x n − . Next,define G n ( x n − ) = q n ( X n ( x n − )) and note G n ( x n − ) ⊂ G n . Now wellorder G n ( x n − ) to have minimal element q n ( x n − ) and otherwise the wellordering of G n ( x n − ) is arbitrary. Next, well order each point preimageof the map q n | X n ( x n − ) : X n ( x n − ) → G n ( x n − ) subject only to theconstraint that x n − = min q n ( x n − ) . To complete the definition of the well ordering on X n suppose weare,given distinctpoints { x n , y n } ⊂ X n . If R n − ( x n ) = R n − ( y n ) werequire the order of X n − to dictate which is bigger. If R n − ( x n ) = R n − ( y n ) and q n ( x n ) = q n ( y n ) we require the order of G n ( R n − ( x n ))to dictate which is bigger. If R n − ( x n ) = R n − ( y n ) and q n ( x n ) = q n ( y n ) we require the order on q ( x n ) to dictate which is bigger. In sum-mary, lexical inspection of the ordered triples ( R n − ( x n ) , q n ( x n ) , x n ) and( R n − ( y n ) , q n ( y n ) , y n ) determines which of { x n , y n } is bigger.Crucially, if { x n , y n , z n } ⊂ X n and x n < y n < z n and q n − R n − ( x n ) = n − R n − ( z n ) , then q n − R n − ( x n ) = q n − R n − ( y n ), and we argue thecontrapositive as follows. Suppose q n − R n − ( x n ) = q n − R n − ( y n ) and q n − R n − ( x n ) = q n − R n − ( z n ) with { x n , y n , z n } ⊂ X n . Let x n − = q n − ( x n ) and y n − = q n − ( y n ) and z n − = q n − ( z n ). By the inductionhypothesis we must have y n − < { x n − , z n − } or y n − > { x n − , z n − } ,and hence, appealing to our local definition, y n < { x n , z n } or y n > { x n , z n } . Finally note R n − ≤ id | X n . To check that axiom 2.1 holds, note the map X n → X ∞ sending x n to ( x , . . . , x n , x n , . . . ) is a topological embedding, henceforth we con-flate the discrete space X n with the corresponding discrete subspace ofeventually constant sequences in X ∞ , the sequences whose terms coin-cide from index n upward. Thus, with moderate abuse of notation, weextend the map R N | X N +1 canonically to R N : X ∞ → X N , so that R N ( x , x , . . . x N , x N +1 , . . . . ) = ( x , x , . . . x N , x N , . . . ), and note R N = R N R N +1 . By definition a point g ∈ G is a subspace g ⊂ X ∞ so that dis-tinct points { x, y } ⊂ g have q n equivalent coordinates for each n , anda point g n ∈ G n is a subspace g n ∈ X n . Our starting assumptionsensure we have a well defined function G n ֒ → G sending g n ∈ G n to( . . . .r n − ( g n ) , g n , j n ( g n ) , j n +1 j n ( g n ) . . . ) ∈ G . This is a topological em-bedding, and henceforth we conflate G n with the corresponding subspaceof G . The map q n R n : X ∞ → G n is constant on sets of the form q − ( g ) and thus there is a unique induced map Π n : G → G n suchthat Π n = Π n Π n +1 . The maps { Π n } determine a continuous injection φ : G → lim ← G n .If φ were a topological embedding then it would follow easily that dimG = 0, since φ embeds G into the 0 dimensional metric space G × G . . . However in general φ is NOT a topological embedding [19].Note each space G n is discrete, since topological quotients of discretespaces are discrete. Moreover G is a quotient of a metrizable space andhence G is sequential. Thus axiom 2.1 will hold provided we can showpointwise convergence Π n → id | G. The latter claim holds by definition,shown as follows. Given g ∈ G and an open U ⊂ G so that g ∈ U ,lift g to some ( x , x , . . . ) ∈ q − ( g ) ⊂ X ∞ and obtain a basic open V ⊂ q − ( U ) ⊂ X ∞ with ( x , x , . . . ) ∈ V so that V = { x } × { x } . . . × { x N } × X N +1 . . . . Now given n ≥ N to check Π n ( g ) ∈ U it suffices to check, since q : X ∞ → G is a quotient map, that some lift of Π n ( g ) ∈ U . Our specialpoint ( x , x , . . . ) ∈ V suffices. Thus axiom 2.1 holds.The space lim ← X n = X ∞ is a topological inverse limit of discretespaces under retraction bonding maps R n | X n +1 . We have well orderedeach set X n so that R n ( x n +1 ) ≤ x n for x n +1 ∈ X n +1 . Thus with theinduced lexigraphic order on X ∞ we have R n ( x ) ≤ x for x ∈ X ∞ . The space X ∞ has the topology of pointwise convergence, and thus,since X n is a discrete space, a sequence { s n } ⊂ X ∞ converges iff Π N ( s n ) iseventually constant for each N . Thus, if the strictly increasing sequence s < s . . . ⊂ X ∞ diverges, then every subsequence of { s n } diverges,since { Π N ( s n ) } is nondecreasing for each N. Conversely, if the increasingsequence { s n } ⊂ X ∞ converges, then, since { Π N ( s n ) } is nondecreasingand eventually constant for each N, lim { s n } = sup { s n } .Since each set X n is well ordered, and since no well ordered set admits strictly decreasing sequence with infinitely many terms, each strictlydecreasing sequence s > s . . . . ⊂ X ∞ converges. Consequently if B ⊂ X ∞ is nonempty and closed, then min( B ) exists. To see why, let b =min Π ( B ). Let b = min Π Π − ( b ) . Let b = min Π Π − ( b ) and so on.Note b ≥ b . . . and let b = lim { b n } . Thus b ∈ B since B is closed .Note b ≤ a for each a ∈ B and thus b = min( B ).Note G is T by Remark 2.4 and in particular G is T . Thus pointpreimages are closed under the map q : X ∞ → G . Define σ : G → X ∞ as σ ( g ) = min( q − ( g )). Since σ is one to one and since subsets of linearlyordered sets are linearly ordered, we obtain a linear order on G definedso that g < h iff σ ( g ) < σ ( h ) . In particular the sequence { g n } is strictlyincreasing in G iff { σ ( g n ) } is strictly increasing in X ∞ . In the processof checking axiom 2.2 we will show σ is left continuous but not rightcontinuous.First we observe some basic properties of our definition of the linearorder on the set X ∞ , conflating X N with the subspace of X ∞ , all of whoseterms are constant from index N and above. If x = ( x , x , . . . . ) ∈ X ∞ then x n ≤ x n +1 ≤ x, and x n → x, and x = sup { x n } . Next we observe a basic property of σ . If ( y , y , . . . . ) = y < z = σ ( g ) = ( z , z , . . . ) ∈ X ∞ and if N is minimal so that y N = z N , then y N The Hawaiian earing HE is a subpace of the plane, theunion of a sequence circles centered at (0 , /n ) with radius /n and sharingthe common point (0 , . The Hawiian earring group G is the fundamentalgroup of HE , the set of path components of the spaced of based loopsin HE , with group operation cancatanation. Endowed with the quotienttopology inherited from the space of based loops in HE , the Hawaiianearring group has large inductive dimension 0.Proof. Let L ( S , , HE, p ) denote the space of based loops in HE. Note L ( S , , HE, p ) is a separable metric space with the uniform metric topol-ogy ( equivalent in this case to the compact open topology), since both S and HE are compact metric spaces. Our plan is to manufacture aspace G as in the hypothesis of Theorem 3.1, and build a quotient map F : L ( S , , HE, p ) → G whose point preimages are precisely the pathcomponents of L ( S , , HE, p ) . Consequently, by basic general topology,there is an induced homeomorphism h : π ( HE, p ) → G , and hence bothspaces have the same dimension.If p ∈ HE is the interesting point, given an aribtrary based loop f ∈ L ( S , , HE, p ), for each component J ⊂ f − ( HE \ p ), we can homo-topically tighten f | ¯ J within its image to a linearly parameterized loop orto a constant. Since f is uniformly continous, the union of the tightenings f wt is continuous and path homotopic to f , and we call the resulting map f wt weak tight. Thus, a loop f wt ∈ L ( S , , HE, p ) is weak tight provided f | J is linear and one to one, for each component J ⊂ f − wt ( HE \ p ). No-tice the weak tight loops W T ( S , , HE, p ) comprise a closed subspace of L ( S , , HE, p ), since being not weak tight is an open property for loopsin L ( S , , HE, p ) . If H denotes the group of orientation preserving homeomorphisms of S which fix 1, then H acts isometrically on W T ( S , , HE, p ) via right com-position, i.e. h ∈ H sends f w ∈ W T ( S , , HE, p ) to the map f w h. Thusby Lemma 4.1 the quotient W T ( S , , HE, p ) /H ( W T ( S , , HE, p )) ismetrizable. For convenience rename the mentioned quotient space X ∞ andthe quotient map Q w : W T ( S , , HE, p ) → X ∞ . A typical point of X ∞ s a weak tight path up to monotone orientation preserving reparameter-ization, i.e. two weak tight paths are equivalent if they pass through thesame points in the same order.Let HE n ⊂ HE denote the bouquet of the first n loops. Thus if wedefine X n ⊂ X ∞ as the subspace with image in HE n we have an inducedretraction R n : X ∞ → X n deleting all large index loops. Crucially notice X n is the discrete monoid on n letters { x , x − , . . . x − n } , with the emptyword corresponding to the constant loop at p , and X ∞ = lim ← X n withbonding map R n | X n +1 . Thus we can think of points of X ∞ as unreducedinfinite words in { x , x − , x ,... } so that each letter appears finitely manytimes.Now let G n denote with the discrete topology, the free group on n letters { x , . . . x n } and let q n : X n → G n denote the canonical quo-tient map. Note the maps { q n } induce an equivalence relation on X ∞ :two infinite words w ∈ X ∞ and v ∈ X ∞ are equivalent iff for all nq n R n ( w )= q n R n ( w ) ∈ G n . Let q : X ∞ → G denote the correspondingquotient map determined by this equivalence relation.The previous paragraphs establish a composition of functions L ( S , , HE, p ) → W T ( S , , HE, p ) → X ∞ → G . The first arrow is a discontinous re-traction, the second and third arrows are continuous quotient maps, andwe let F denote the composition L ( S , , HE, p ) → G. By definition π ( HE, p ) is the quotient of L ( S , , HE, p ) modding out by the pathcomponents. Thus, to prove the existence of an induced homeomorphism h : π ( HE, p ) → G , it suffices, by basic general topology, to show that F is a quotient map whose point preimages are the path components of L ( S , , HE, p ) . Let W : L ( S , , HE, p ) → W T ( S , , HE, p ) denote thediscontinuous retraction described previously.To check continuity of F suppose f n → f uniformly in L ( S , , HE, p ).We apply Lemma 4.4 to the sequence { F ( f n ) } and first show { Π N ( F ( f n )) } is eventually constant for each N . Let κ N : HE → HE N denote thecanonical retraction, notice locally at f, the composition { κ N ( f n ) } even-tually preserves the homotopy path class of { κ N ( f ) } in HE N . Thus { Π N ( F ( f n )) } is eventually constant. To check that { σ ( F ( f n )) } has com-pact closure we will apply Ascoli’s Theorem. First note the map W pre-serves or improves equicontinity data (and the image of 1 ∈ S is con-stant and thus convergent), and hence { W ( f n ) } has compact closure in W T ( S , , HE, p ).The following definition has an algebraic analogue, the different waysthat one might start with an unreduced word in X N and then cancelinverse pairs to create the irreducible representive. Given a weak tightloop β ∈ W T ( S , , HE, p ) and a natural number N , define Σ( β, N ) ⊂ W T ( S , , HE, p ) as the subspace of irreducicle loops in HE N , obtainedby starting with κ N ( β ) and deleting successive nonconstant p based inessen-tial loops, replacing each with the constant map p. The important observation is that each loop in Σ( β, N ) has equicon-tinuity data no worse than that of β. Thus, since { W ( f n ) } has compactclosure, the union over N and n of the subspaces Σ( W ( f n ) , N ) has com-pact closure in W T ( S , , HE, p ). Call the latter compactum C , recallLemma 4.4 and observe { σF ( f n ) } ⊂ C . Thus F is continous by Lemma4.4. Since qQ w is a quotient map, and since the (discontinous) map W is retraction, it follows from Lemma 4.2 that F is a quotient map.To see that point preimages of F are precisely the path componentsof L ( S , , HE, p ) , first note W ( β ) is path homotopic in HE to β . Thusif α and β are path homotopic in HE then W ( α ) and W ( β ) are pathhomotopic in HE . Thus q n R n ( Q w W ( α )) = q n R n ( Q w W ( β )) for all n andhence F ( α ) = F ( β ) . Conversely, since the Hawaiian earring is π shapeinjective, if α and β are not path homtopic in HE then q n R n ( Q w W ( α )) = q n R n ( Q w W ( β )) ofr some n and hence F ( α ) = F ( β ) . Thus π ( HE, p ) withthe quotient topology, is homeomorphic to G . It follows from Theorem3.1 that G has large inductive dimension zero, and hence π ( HE, p ) haslarge inductive dimension zero. Lemma 4.1. Suppose ( X, d ) is a metric space and H is a group (underfunction composition) of isometries of X. Then the orbit closures underthe action forms a partition of X and the Hausdorff metric is compatiblewith the quotient topology. (It is not necessary to assume X is complete,that the orbits are bounded, or that the action is free.) Given x ∈ X define C ( x ) = { H ( x ) } . Thus C ( x ) is a typical element of the quotient space.With moderate abuse of notation we denote the quotient space X/H ( X ) ).Proof. Given x ∈ X define C ( x ) = { H ( x ) } . To check the orbit closures aredisjoint, given y ∈ C ( x ) let y = limh n ( x ) for some sequence { h n } ⊂ H .Suppose ǫ > z ∈ H ( y ). Let z = h ( y ), get n so that d ( y, h n ( x )) < ǫ .Then d ( z, hh n ( x )) < ǫ . Thus z ∈ C ( x ) and hence H ( y ) ⊂ C ( x ). Thus C ( y ) ⊂ C ( x )since C ( x ) is closed. By a symmetric argument C ( x ) ⊂ C ( y ) and thus C ( x ) = C ( y ) . Thus the sets of the form C ( x ) determine apartition of X into pairwise disjoint closed sets.Given orbit closures C ( x ) and C ( y ) let ε denote inf { d ( x, y ) } takenover all z ∈ H ( y ). Define D ( C ( x ) , C ( y )) = ε . It is straight forward tocheck this is the Hausdorff metric, and the canonical map X → X/H ( X )is a contraction. To check it’s a quotient map. Suppose A ⊂ X/H ( X ) isnot closed with C ( a n ) → C ( x ) with C ( a n ) ∈ A and C ( x ) / ∈ A . Obtain x n ∈ C ( a n ) with x n → x . Thus the preimage of A is not closed in X . Lemma 4.2. Suppose X is a space and r : X → Y is a (possibly disconti-nous) retraction onto the subspace Y . Suppose q : Y → Z is a (continuous) quotient map and A ⊂ Z is not closed. Then ( qr ) − ( A ) is not closed inX.Proof. Pullback A to B = q − ( A ) ⊂ Y. Note B is not closed in Y since q is a quotient map. Thus B = ( qr ) − ( A ) is not closed in X. Lemma 4.3. Suppose the p based loops f n and g n are path homotopicin the Hawaiian earring HE with p the special point. Suppose f n → f uniformly and g n → g uniformly. Then f and g are path homotopic. roof. Cancatanating with the reverse path, the inessential loops f n g − n → fg − uniformly. Since the bouquet of n loops HE n is locally contractible, fg − is inessential in HE n for each n and hence, since HE is π shapeinjective fg − is inessential. See also a direct proof in [15] Lemma 4.4. Suppose X n is the discrete free monoid on letters { x , x − , x , . . . .x − n } with empty identity. Suppose R n | X n +1 → X n is the forgetful retraction,deleting all occurences of { x n , x − n } . Let X ∞ = lim ← X n . Identify X n with the subspace of X ∞ comprised of all sequences eventually constantfrom index n onward. Let R n : X ∞ → X n denote the canonical retraction.Let q n : X n → G n denote the canonical quotient onto the free group G n on n letters. Let σ n : G n → X n denote the embedding mapping g n ∈ G n toits maximally reduced representative. Let q : X ∞ → G denote the quotientmap under the equivalence relation u ∼ v iff q n R n ( u ) = q n R n ( v ) for all n. By definition q n R n descends to the quotient inducing a map Π n : G → G n . Claim 1. There is a well defined (discontinuous) injection σ : G → X ∞ with σ ( g ) = lim n →∞ σ n Π n ( g ) and σ ( g ) ∈ g. Claim 2. The sequence { g n } converges in the space G iff for all N the sequence Π N ( g n ) is eventuallyconstant, and also if the sequence { σ ( g n ) } has compact closure.Proof. We have a canonical partial order on X ∞ defined as follows. Given w = ( w , w , . . . . ) ∈ X ∞ let T ( w , w , . . . ) = ( N ( w ) , N ( w ) , . . . ) with N k ( w ) ≥ { x k , x − k } in w k . Thefunction T determines a partial lexigraphic order on X ∞ with T ( v ) 9] R. Brown, Ten topologies for X × Y , Q. J. Math. Oxf. 14 (1963)303–319[10] Cannon, J.W., Conner, G.R., The combinatorial structure of theHawaiian earring group, Topology and its Applications 106 (2000),225–271.[11] G.R. Conner, K. Eda, Fundamental groups having the whole infor-mation of spaces, Topology Appl. 146 (147) (2005) 317–328.[12] E. Cheng, x+y, A Mathematician’s Manifesto for Rethinking Gender,June 2020, New York, NY, Basic Books.[13] Conner, G.R., Kent, C.: Fundamental groups of locally connectedsubsets of the plane. Adv. Math. 347, 384–407 (2019)[14] Corson, S., The number of homomorphisms from the Hawaiian ear-ring group. J. Algebra 523, 34–52 (2019)[15] B. de Smit, The fundamental group of the Hawaiian earring is notfree, Internat. J. Algebra Comput. 2 (1) (1992)[16] J. J. Dijkstra and J. van Mill, Erd˝os space and homeomorphismgroups of manifolds, Mem. Amer. Math. Soc. 208 (2010), no. 979,vi+62.[17] K. Eda, Free subgroups of the fundamental group of the Hawaiianearring, J. Algebra 219 (1999), 598–605.[18] K. Eda, Atomic property of the fundamental groups of the Hawaiianearring and wild locally path-connected spaces, J. Math. Soc. Japan63 (2011) 769–787.[19] P. Fabel, The topological Hawaiian earring group does not embedin the inverse limit of free groups, Algebr. Geom. Topol. 5 (2005),1585–1587.[20] P. Fabel, Multiplication is discontinuous in the Hawaiian earringgroup (with the quotient topology), Bull. Pol. Acad. Sci., Math. 59(1) (2011) 77–83.[21] P. Fabel, Compactly generated quasitopological homotopy groupswith discontinuous multiplication, Topol. Proc. 40 (2012) 303–309.[22] H. Fischer, A. Zastrow, Generalized universal covering spaces andthe shape group, Fundam. Math. 197 (2007)ˇZ.[23] H. Fischer, A. Zastrow, A core-free semicovering of the HawaiianEarring, Topology Appl. 160 (14) (2013) 1957–1967.[24] J.W. Morgan, I. Morrison, A Van Kampen theorem for weak joins,Proc. London Math. Soc. 53 (3) (1986) 562–576.[25] H. Porst, On the existence and structure of free topological groups,in: Category Theory at Work, 1991, pp. 165–176.[26] Z. Virk, A. Zastrow, The comparison of topologies related to variousconcepts of generalized covering spaces, Topol. Appl. 170 (2014) 52–62.[27] Z. Virk, A. Zastrow, A new topology on the universal path space,Topol. Appl. 231 (2017) 186–196., Q. J. Math. Oxf. 14 (1963)303–319[10] Cannon, J.W., Conner, G.R., The combinatorial structure of theHawaiian earring group, Topology and its Applications 106 (2000),225–271.[11] G.R. Conner, K. Eda, Fundamental groups having the whole infor-mation of spaces, Topology Appl. 146 (147) (2005) 317–328.[12] E. Cheng, x+y, A Mathematician’s Manifesto for Rethinking Gender,June 2020, New York, NY, Basic Books.[13] Conner, G.R., Kent, C.: Fundamental groups of locally connectedsubsets of the plane. Adv. Math. 347, 384–407 (2019)[14] Corson, S., The number of homomorphisms from the Hawaiian ear-ring group. J. Algebra 523, 34–52 (2019)[15] B. de Smit, The fundamental group of the Hawaiian earring is notfree, Internat. J. Algebra Comput. 2 (1) (1992)[16] J. J. Dijkstra and J. van Mill, Erd˝os space and homeomorphismgroups of manifolds, Mem. Amer. Math. Soc. 208 (2010), no. 979,vi+62.[17] K. Eda, Free subgroups of the fundamental group of the Hawaiianearring, J. Algebra 219 (1999), 598–605.[18] K. Eda, Atomic property of the fundamental groups of the Hawaiianearring and wild locally path-connected spaces, J. Math. Soc. Japan63 (2011) 769–787.[19] P. Fabel, The topological Hawaiian earring group does not embedin the inverse limit of free groups, Algebr. Geom. Topol. 5 (2005),1585–1587.[20] P. Fabel, Multiplication is discontinuous in the Hawaiian earringgroup (with the quotient topology), Bull. Pol. Acad. Sci., Math. 59(1) (2011) 77–83.[21] P. Fabel, Compactly generated quasitopological homotopy groupswith discontinuous multiplication, Topol. Proc. 40 (2012) 303–309.[22] H. Fischer, A. Zastrow, Generalized universal covering spaces andthe shape group, Fundam. Math. 197 (2007)ˇZ.[23] H. Fischer, A. Zastrow, A core-free semicovering of the HawaiianEarring, Topology Appl. 160 (14) (2013) 1957–1967.[24] J.W. Morgan, I. Morrison, A Van Kampen theorem for weak joins,Proc. London Math. Soc. 53 (3) (1986) 562–576.[25] H. Porst, On the existence and structure of free topological groups,in: Category Theory at Work, 1991, pp. 165–176.[26] Z. Virk, A. Zastrow, The comparison of topologies related to variousconcepts of generalized covering spaces, Topol. Appl. 170 (2014) 52–62.[27] Z. Virk, A. Zastrow, A new topology on the universal path space,Topol. Appl. 231 (2017) 186–196.