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The Hawaiian earring group is topologically incomplete

The premier exhibition of the following phenomenon: The fundamental group of any Peano continuum constructed in similar fashion to the Hawaiian earring admits two natural distinct topological group structures. However despite being uncountable and regular, neither group is a Baire space and hence neither group admits a compatible complete metric.


Introduction
Endowed with the quotient topology from the space of based loops in X, the familiar fundamental group π 1 (X, p) of a space X becomes a topological group. As a subject in its own right, the theory of topological fundamental groups is in the early stages of development, and the extent to which the theory will prove useful remains to be seen. For example, the theory has no extra value in the context of locally contractible spaces, since if X is locally contractible, then π 1 (X, p) inherits the discrete topology ([11]) and encodes no more information then algebraic data.
On the other hand, if we also consider spaces that are not locally contractible then the theory begins to earn its keep. For example both the algebra and topology of π 1 (X, p) is invariant under the homotopy type of the underlying space (Corollary 3.4 [1]). Consequently, by analyzing the topology of π 1 (X, p), the theory has the capacity to distinguish spaces of distinct homotopy types when the familiar algebraic homotopy groups fail to do so (see [13] for examples.) In any mathematical theory (e.g. the theory of groups, the theory of n manifolds, the theory of Banach spaces, etc...) the investigation of reasonable conjectures tends to work in tandem with the cataloging of properties of the fundamental examples. This paper was written with the latter goal in mind. In particular we investigate the topological fundamental group of one of the simplest nonlocally contractible spaces, the Hawaiian earring HE. The space HE is the union of a null sequence of simple closed curves joined at a common point p. Hence HE is a one dimensional metric Peano continuum, and the deletion of one point leaves a (disconnected) one manifold.
The following tool is indispensable for the investigation of π 1 (HE, p). Consider HE = lim ← X n as the inverse limit of a nested sequence of retracts X 1 ⊂ X 2 , ... where X n is the union of n simple closed curves joined at a common point p, and consider the canonical homomorphism φ : π 1 (HE, p) → lim ← π 1 (X n , p). Since φ is continuous (Proposition 3.3 [1]), and since X n is locally contractible, the question of whether π 1 (HE, p) is a T 1 space (a space whose one point subsets are closed) is precisely equivalent (see [14]) to the question of whether φ is one to one. A comprehensive affirmative answer to the latter question constitutes the lion's share of a paper of Morgan and Morrison [17].
Since the T 1 space π 1 (HE, p) is also a topological group, it follows from general properties of topological groups that π 1 (HE, p) is in fact completely regular. It is an open question whether π 1 (HE, p) is normal or whether π 1 (HE, p) is metrizable.
Having said this we should point out that the metrizabilty (and hence normality) of π 1 (HE, p) would follow from the published work of another author, work that this author believes to be incorrect. A lack of clarity makes it difficult to pinpoint the location of the first mistake. The paper [1] seems to claim that φ : π 1 (HE, p) → lim ← π 1 (X n , p) is an isomorphism with continuous inverse (Theorem 8.1 [1]). Surjectivity is easily seen to fail since, in lim ← π 1 (X n , p), the "infinite word" .. has no counterpart in π 1 (HE, p). The failure of φ to be a homeomorphism onto its image is not so easy to see, and an explicit counterexample is contained in my paper [10].
Fortunately the main results of this paper do not depend on these contestable claims. We prove that if X is the Hawaiian earring, or if X is constructed in similar fashion to the Hawaiian earring, then π 1 (X, p), despite being uncountable, is not a Baire space.
An immediate application is the following: Ignoring the question of whether the regular space π 1 (X, p) is metrizable, we can conclude by the Baire category theorem that π 1 (X, p) admits no complete metric compatible with its topology. This in turn yields a 'no retraction theorem' whose roots depend on the topological rather than the algebraic properties of the groups in question: Because π 1 (X, p) is not a Baire space, the main result of [12] shows that X cannot be embedded as retract of any space Y such that π 1 (Y, p) admits a complete metric.
Finally, in the process of proving π 1 (X, p) is not a Baire space we also prove that ιm(φ) is not a Baire space in lim ← π 1 (X n , p). Thus this paper seems to contain the premier exhibition of the following phenomenon: There exists a Peano continuum X such that with either of two natural but distinct T 1 topological group structures, the fundamental group of X admits no complete metric compatible with its topology.
It is hoped that uncovering the properties of the simplest examples such as HE will serve as a useful benchmark when pursuing further developments in the theory of topological fundamental groups.

Preliminaries
Given a set S a word in S is either the empty set or a finite sequence in S. The entries of a word are called letters. A subword v of the word w is the word obtained by deleting finitely may letters from w. Suppose . is a sequence (either finite or infinite) of pairwise disjoint groups. Let W denote the set of words in the set To define K(w) first partition w into a minimal number of cells such that each cell is a subword of w consisting of consecutive letters of w, such that each letter is of the same type. Delete each cell whose multiply to make id H i . Replace the remaining cells by the product of its letters. Call the resulting subword K(w). Notice K(w) has no more letters than w, and that K(w) and w have the same number of letters if and only if K(w) = w. Thus the sequence w, K(w), K 2 (w), ... is eventually constant. If K(w) = w we say w is reduced. Moreover a word w is reduced if and only if w is empty word or w is a nonempty word in the set (H 1 ∪ H 2 ∪ ..)\{id H 1 , id H 2 , ..} such that no two consecutive letters are of the same type.
Define RD : W → W such that RD(w) = lim n→∞ K n (w). Declare two words w and v in W equivalent if RD(v) = RD(w). By construction each equivalence class contains a unique reduced word.
A topological space Z is a Baire space provided ∩ ∞ n=1 U n is dense in Z whenever each of U 1 , U 2 , .. is an open dense subspace of Z. It is a classical fact that every complete metric space is a Baire space (Theorem 7.2 [18]).

Definitions
For the remainder of the paper we make the following assumptions.
All topological fundamental groups are endowed with the quotient topology.
Suppose Xˆ= ∪ ∞ n=1 X n is a metrizable topological space. Assume X 1 ⊂ X 2 ⊂ X 3 .. is a nested sequence of locally simply connected subspaces with p ∈ X 1 .
Assume X n+1 = X n ∪ Y n with X n ∩ Y n = p and assume Y n is not simply connected.
Let r n : X n+1 → X n denote the retraction collapsing Y n+1 to the point p.
Assume Xˆadmits a topologically compatible metric such that diam(Y n ) → 0. This assumption serves, for example, to distinguish the (compact) Hawaiian earring from the (noncompact) union of countably many circles of radius 1 joined at the common point p.
Let X = lim ← X n , the inverse limit space under the maps r n . Let R n : X → X n denote the map R n (x 1 , x 2 , ...) = x n Let φ : Let V denote the set of all words in ∪ ∞ n=1 H n . Words in V are allowed to be not reduced. Define Let ψ n : G n+1 → G n denote the canonical epimorphism ψ n = RD(κ n ). Thus, to compute ψ n of the reduced word w n+1 , first delete from w n+1 all letters of type n + 1, and then reduce the resulting subword as much as possible.
Let G denote the subgroup of lim ← G n consisting of all sequences (w 1 , w 2 , ...) such that for each m ≥ 1 the following sequence is eventually constant: 4 Facts about the preliminary data Proposition 1 Xˆis canonically homeomorphic to X under the map h : ∪ ∞ n=1 X n → X satisfying h(x n ) = (p, p, , , x n , x n , ..). Moreover h Xn ֒→ X is an embedding. Henceforth we consider Xˆand X to be the same space.

Proposition 2
The van Kampen theorem determines a canonical isomorphism j * n : π 1 (X n , p) → G n . Henceforth we will consider these groups to be the same. See [16] for details.
Most of [17] is devoted to Theorem 4.1, which, translated into the notation of this paper, becomes Proposition 3.
The following proposition is perhaps surprising and is the subject of [10].
5 Main result: π 1 (X, p) is not a Baire space Lemma 5 For each N ≥ 1, the set σ −1 (N) is closed in lim ← G n .
Next we show that at least one of the following inequalities hold: ..x ks . Thus, for each n at least one of f n or f n does not belong to σ −1 (K). Remark 8 It is shown in [10] that φ is not an embedding. Hence, considering example 7, Lemma 9 is not an immediate consequence of Lemma 6. However the same idea drives both proofs. Proof. Since Y n is not simply connected for each n ≥ 1 select h n ∈ C p (X n ) such that [h n ] ∈ π 1 (Y n, p)\id n . Let h ∞ denote the constant map p. Let U = φ −1 (σ −1 (K)). Let F : C p (X) → π 1 (X, p) denote the canonical surjective quotient map. Let A ⊂ U. Let B = F −1 (U). Since F is a quotient map, in order to prove A is not open it suffices to prove B is not open in C p (X). Suppose [α] ∈ A. For each n ∈ {∞}∪{1, 2, 3, ...} let g n = h 1 * h n * h −1 1 * h −1 n * α denote the path in C p (X) obtained by concatenation and compatible with the partition {0, 1 5 , 2 5 , 3 5 , 4 5 , 1}. In similar fashion define g n = h n * h 1 * h −1 n * h −1 1 * α. Note g n → g ∞ and g n → g ∞ uniformly and {g ∞ , g ∞ } ⊂ B since g ∞ and g ∞ are path homotopic in X to α. For n ≥ 2 let f n = φF (g n ), let f n = φF (g n ) and let f = φF (α). Since φF is continuous, we conclude f n → f and f n → f. Now we may appeal to the proof of Lemma 6 to conclude that if n ≥ K + 1 then at least one f n or f n is not an element of σ −1 (K). Thus at least one of g n or g n does not belong to B. Hence B is not open and this proves Lemma 9.
Theorem 10 The topological groups G and π 1 (X, p) are not Baire spaces.
Proof. To prove that a topological space Z is note a Baire space it suffices to prove Z is the countable union of closed subspaces Z 1 ∪ Z 2 ...such that Z n has empty interior for each n. Taking Z n = σ −1 (n) it follows directly from Lemmas 5 and 6 that G is not a Baire space. To prove π 1 (X, p) is not a Baire space consider the sets A n = φ −1 (Z n ). Since φ is continuous, A n is closed by Lemma 5. Lemma 9 shows A n has empty interior.
Corollary 11 Endowed with either the quotient topology or the inverse limit space topology, π 1 (X, p) does not admit a complete metric.