Higher Dimensional Earrings

Posted on November 23, 2019 by Jeremy Brazas

2d Hawaiian Earring 03

The (1-dimensional) earring space \mathbb{E} is a 1-dimensional Peano continuum (connected, locally path-connected, compact metric space) constructed by adjoining a shrinking sequence of circles at a single point.

Hawaiian earring space

The earring space

The importance of \mathbb{E} stems from the fact that this is the prototypical space that has infinite products in it’s fundamental group. This makes it a useful test-space for detecting the existence of non-trivial infinite products (or algebraic wildness) in other fundamental groups. The earring group is not a free group but it is basically an infinitary (i.e. having infinite products) analogue of a free group. Moreover, \mathbb{E} is aspherical, i.e. \pi_k(\mathbb{E},b_0)=0 for k\geq 2. In this post, I’m going to give a summary about their higher dimensional analogues \mathbb{E}_n\subseteq \mathbb{R}^{n+1}, the n-dimensional earring.

Constructing \mathbb{E}_n

Definition: let n\geq 0 and C_{k}^{n}=\{\mathbf{x}\in\mathbb{R}^{n+1}\mid \|\mathbf{x}-(1/k,0,0,\dots ,0)\|=1/k\} be the n-sphere of radius 1/k centered at (1/k,0,0,\dots,0). The n-dimensional earring is the set

\mathbb{E}_n=\bigcup_{k\in\mathbb{N}}C_{k}^{n}

with the subspace topology inherited from \mathbb{R}^{n+1}. Let b_0=(0,0,\dots,0) denote the origin, which we typically take as the basepoint.

2dim

Two views of \mathbb{E}_2, the 2-dimensional earring

The 0-dimensional earring \mathbb{E}_0=\{0,\dots,1/3,2/5,1/2,2/3,1,2\} is simply a convergent sequence and \mathbb{E}=\mathbb{E}_1 is the usual earring. We can build \mathbb{E}_n using a variety of constructions:

  1. One-point compactification: \mathbb{E}_n is homeomorphic to the one-point compactification of a countable disjoint union of open n-cells \coprod_{k\in\mathbb{N}}(0,1)^n, i.e. \mathbb{E}_n\cong \coprod_{k\in\mathbb{N}}[0,1]^n/\coprod_{k\in\mathbb{N}}\partial [0,1]^n.
  2. Subspace of a product: \mathbb{E}_n is homeomorphic to the wedge sum \bigvee_{k\in\mathbb{N}}S^n viewed naturally as a subspace of \prod_{k\in\mathbb{N}}S^n with the product topology. To make this identification, the m-th wedge summand is identified with the set \prod_{k\in\mathbb{N}}A_k where A_k=\{b_0\} if k\neq m and A_m=S^n.
  3. Inverse Limit: Let X_k=\bigcup_{i=1}^{k}C^{n}_{i} be the union of the first k n-spheres and r_{k+1,k}:X_{k+1}\to X_k be the retraction that collapses C_{k+1}^{n} to b_0 and is the identity elsewhere. Then the natural retractions r_k:\mathbb{E}_n\to X_k collapsing \bigcup_{i>k}C_{i}^{n} to b_0 induce a homeomorphism \mathbb{E}\cong\varprojlim_{k}(X_k,r_{k+1,k}).
  4. Reduced Suspension: For n\geq 1, the n-dimensional earring \mathbb{E}_{n} is homeomorphic to the reduced suspension \Sigma \mathbb{E}_{n-1} of the (n-1)-dimensional earring. By iteration, we obtain a formula similar to that for ordinary spheres: \mathbb{E}_{n+m}\cong\Sigma^m\mathbb{E}_n for all m,n\geq 0. It is important here that we use the reduced suspension and not the unreduced suspension. The undreduced suspension S\mathbb{E}_{n-1} is not homotopy equivalent to \Sigma \mathbb{E}_{n-1}. This means that your typical tricks, including Mayer Vietoris Sequences, won’t be helpful for computing the singular homology groups of \mathbb{E}_{n}.

The 2-dimensional earring \mathbb{E}_2, was shown to have some interesting properties in the famous paper [1] by Barratt and Milnor. To put their observation into context, recall that if we consider an ordinary wedge of 2-spheres, elementary computations show that

\displaystyle\widetilde{H}_k\left(\bigvee_{\mathbb{N}}S^2\right)=\begin{cases} \bigoplus_{\mathbb{N}}\mathbb{Z} & k=2 \\ 0 & k\neq 2 \end{cases}.

It might seem that \mathbb{E}_2 should be similar. In particular, \mathbb{E}_2 is simply connected and locally simply connected. However, Barratt and Milnor showed that there are infinitely many k>2 such that H_k(\mathbb{E}_2) is uncountable! This seems really strange and “anomalous” until one reads the paper to see how it’s done.

I’ve said it once and I’ll say it again: the “wildness” in wild algebraic topology is really just about infinite products. The reason why higher homology groups of \mathbb{E}_2 are often non-trivial is precisely because one can consider infinite sums \sum_{m=1}^{\infty}[f_m] of elements of [f_m]\in \pi_k(\mathbb{E}_2), e.g. given by shrinking sequences of non-trival Whitehead products. Even though [f_m] is trivial in homology of any finite wedge of the spheres C_{k}^{2}, the k-cycle represented by the infinite \sum_{m=1}^{\infty}[f_m] (apply the Hurewicz homomorphism) is not the boundary of a finite sum of n+1-chains and therefore represents a non-trivial homology class.

The homotopy groups \pi_k(\mathbb{E}_n), k\leq n

Recall that a space (X,x_0) is n-connected if \pi_m(X,x_0) is trivial for 0\leq m\leq n.

Theorem (Eda-Kawamura): \mathbb{E}_n is (n-1)-connected and locally (n-1)-connected. Moreover, the embedding \varphi:\mathbb{E}_n\to \prod_{k=1}^{\infty}C_{k}^{n} induces an isomorphism on \pi_n.

This means that

\pi_n(\mathbb{E}_n,b_0)\cong \prod_{k=1}^{\infty}\mathbb{Z}

is the Baer-Specker group.

Easy part of the proof: \varphi_{\#}:\pi_n(\mathbb{E}_n,b_0)\to\prod_{k=1}^{\infty}\mathbb{Z} is onto.

Identify \pi_n(C_{k}^{n},b_0) with \mathbb{Z}. Given a sequence of integers (a_1,a_2,a_3,\dots)\in\prod_{k=1}^{\infty}\mathbb{Z}, let f_k:(I^n,\partial I^n)\to (C_{k}^{n},b_0) be a map such that [f_k]=a_k\in \pi_n(C_{k}^{n},b_0). Define f=\prod_{k=1}^{\infty}f_k as the infinite concatenation of n-loops. Explicitly, f:(I^n,\partial I^n)\to (\mathbb{E}_n,b_0) is defined so that the restriction f:\left[\frac{k-1}{k},\frac{k}{k+1}\right]\times I^{n-1}\to\mathbb{E}_n is f(t_1,t_2,\dots, t_n)=f_k((1 + k) (1 + k (t_1-1)),t_2,t_3,\dots,t_n), for all k\geq 1 and f(1,t_2,t_3,\dots,t_n)=b_0.

higherinfproduct

The map f as an infinite concatenation of 2-loops

It’s now easy to see that \varphi_{\#}([f])=(a_1,a_2,a_3,\dots ), proving that \varphi_{\#} is surjective.

Hard part of the proof: \varphi_{\#} is injective.

The Eda-Kawamura proof in [2] is inspired but is written in a very technical fashion and has scared off many-a-reader. Essentially, the key idea of the proof is to simultaneously perform simplicial/cellular approximation on an infinite sequence of disjoint polyhedral domains in I^n. Then you work to perform an infinite Eckmann-Hilton type argument, to show that every map (I^n,\partial I^n)\to (\mathbb{E}_n,b_0) is homotopic to one of the form f=\prod_{k=1}^{\infty}f_k as described above, i.e. where f_k has image in C_{k}^{n}. Then if \varphi_{\#}([f])=(0,0,0,\dots), each f_k has degree 0 as a map S^n\to C_{k}^{n} and thus is null-homotopic in C_{k}^{n}. Let H_k:I^n\times I\to C_{k}^{n} be such a null-homotopy where H_{k}(\mathbf{t},0)=f_k(\mathbf{t}) and H_k(\mathbf{t},1)=b_0. Then we can define a null-homotopy H:I^n\times I\to \mathbb{E}_n of f so that H(\mathbf{t},s)=H_k(\mathbf{t},(1 + k) (1 + k (s-1))) for s\in \left[\frac{k-1}{k},\frac{k}{k+1}\right] and H(\mathbf{t},1)=b_0.

I do wonder if there is a simpler inverse limit type approach to proving that \varphi_{\#} is injective where the null-homotopy H=\varprojlim_{k}G_k is built as an inverse limit of null-homotopies G_k:I^n\times I\to \bigcup_{j\geq k}C_{j}^{n}. It would still be highly non-trivial, but would lend some insight into dealing with higher dimensional homotopy groups. If possible, one would likely have to inductively build the maps G_k by applying simplicial approximation at each step.

The homotopy groups \pi_k(\mathbb{E}_n), k> n

As noted in my problem list, these groups are still unknown. Of course, the higher homotopy groups of spheres are mysterious themselves here so the ultimate goal would be to express \pi_k(\mathbb{E}_n) in terms of homotopy groups of spheres.

From above, we may consider \mathbb{E}_n as a closed subspace of the direct product (S^n)^{\infty}=\prod_{\mathbb{N}}S^n over the naturals. The key is to notice that the argument for the “easy” part of the proof above applies to any dimension: for any k\in\mathbb{N}, the inclusion \mathbb{E}_n\to (S^n)^{\infty} induces a surjective homomorphism \varphi_{\#}:\pi_k(\mathbb{E}_n)\twoheadrightarrow \pi_k((S^n)^{\infty})=\prod_{\mathbb{N}}\pi_k(S^n). Moreover, by applying infinite commutativity, i.e. infinite Eckmann-Hilton, this surjection is a split epi.

Let’s look at the homotopy long exact sequence of the pair ((S^n)^{\infty},\mathbb{E}_n):

up5

The surjections tells us that we really have

up6

which splits up into short exact sequences:

up7

Since the epi \varphi_{\#}:\pi_k(\mathbb{E}_n)\twoheadrightarrow\prod_{k=1}^{\infty}\pi_k(S^n) always splits canonically, we have

\pi_k(\mathbb{E}_n)\cong \pi_{k+1}((S^n)^{\infty},\mathbb{E}_n)\oplus \prod_{\mathbb{N}}\pi_k(S^n)

This tells us at least that explicitly computing \pi_k(\mathbb{E}_n) requires computing \pi_k(S^n). Notice that the Eda-Kawamura computation only tells us that \pi_{n+1}((S^n)^{\infty},\mathbb{E}_n)=0.

An interesting case is

\pi_{3}(\mathbb{E}_2)\cong \pi_{4}((S^2)^{\infty},\mathbb{E}_2)\oplus \prod_{\mathbb{N}}\pi_{3}(S^2)\cong \pi_{4}((S^2)^{\infty},\mathbb{E}_2)\oplus \prod_{\mathbb{N}}\mathbb{Z}

since \pi_{3}(S^2)\cong \mathbb{Z} is generated by the Hopf map.

So the cheesesteak-worthy question is: what is \pi_{k+1}((S^n)^{\infty},\mathbb{E}_n) when k>n?

It will typically be non-trivial because of Whitehead products and infinite products of Whitehead products…just like in Barratt-Milnor! I have my own ideas about what the answer is in the case k=n+1, but there is one technical hurdle in the way from making it precise. There is a piece of technology missing that needs to be developed.

References:

[1] M.G. Barratt, J. Milnor, An example of anomalous singular theory, Proc. Amer. Math. Soc. 13 (1962) 293-297.

[2] K. Eda, K. Kawamura, Homotopy and Homology Groups of the n-dimensional Hawaiian earring, Fund. Math. 165 (2000) 17-28.

Added 3-12-20. I recently learned that the splitting of the exact sequence above appears in the following:

[3] K. Kawamura, Low dimensional homotopy groups of suspensions of the Hawaiian earring, Colloq. Math. 96 (2003) no. 1 27-39.

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New Problem List

Posted on September 8, 2019 by Jeremy Brazas

Check out the new Problem List page!

Do you know of any problems I should add?

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Topological wildness vs. algebraic wildness

Posted on August 14, 2019 by Jeremy Brazas

What does “wildness” really refer to?

I’ll post about higher dimensional wildness soon but to avoid getting too general, I’m going to focus on a more specific question: what does it mean for a space to have a wild fundamental group?

If you surveyed mathematicians with this question most responses would likely say:

X has a wild fundamental group if X is not semilocally simply connected.

You might get a few answers of “not locally contractible,” which is a good answer to a more general question but that would be a deflection from our current focus on \pi_1. A space can easily fail to be locally contractible but still have tame 1-dimensional homotopy. The first answer is reasonable because we often study fundamental groups using covering space theory and, assuming locally path-connectivity, X has a universal (i.e. simply connected) covering space if and only if X is semilocally simply connected. But does failing to be semilocally simply connected really imply that the topology interacts with the algebra of \pi_1 in any kind of wild fashion?

The answer is: a lot of the time, but not always.

I’ll detail both parts of this answer but I think the not always part of the answer involves some fun topology. First, some quick and possibly skip-able review:

Definition: A space X is semilocally simply connected at x\in X if there exists an open neighborhood U such that the inclusion U\to X induces the trivial homomorphism \pi_1(U,x)\to \pi_1(X,x) on fundamental groups. We say X is semilocally simply connected if it is semilocally simply connected at all of its points.

Intuition: Despite the complicated name, the idea behind being “semilocally simply connected” is pretty straightforward. The triviality of the homomorphism \pi_1(U,x)\to \pi_1(X,x) simply means that every loop in U based at x can be contracted in X (but not necessarily in U). For instance, let X\subset\mathbb{R}^3 be a cone of height 1 over a circle in the xy-plane. Then the open set U=X\cap (\mathbb{R}^2\times [0,1/2)) is a annular neighborhood of the base circle and is therefore homotopy equivalent to a circle. Any loop in U is null homotopic in X but a loop going around the base circle once is non-trivial in U since you must use the top of the cone to contract it. In this case, \pi_1(U,x)\to \pi_1(X,x) is trivial even though \pi_1(U,x) is not.

Again, this property is a key assumption in classical covering space theory: If X is path connected and locally path connected, then X has a simply connected (i.e universal) covering space if and only if X is semilocally simply connected.

Let’s analyze the situation where a space fails to have this property.

Topological Wildness

Definition: The topological 1-wild set of X is the subspace

\mathbf{w}(X)=\{x\in X\mid X\text{ is not semilocally simply connected at }x\}.

Hence, for locally path-connected X, the set \mathbf{w}(X) is precisely the topological obstruction to the existence of a universal covering space. If X isn’t locally path connected, remember that we have this handy dandy tool.

Examples:

  • \mathbf{w}(X)=\emptyset if and only if X is semilocally simply connected. Included are all simply connected spaces, CW-complexes, polyhedra, and manifolds.
  • If \mathbb{H} is the Hawaiian earring with join point b_0, then \mathbf{w}(\mathbb{H})=\{b_0\} is a singleton.
  • The space pictured below has a shrinking sequence of Hawaiian earrings attached to the points \frac{1}{2^{n-1}}\in [0,1]. The wild set is \{0,\dots,1/16,1/8,1/4,1/2,1\}, which is not discrete.wildsetNotice that this space is wild at 0 because every neighborhood of that point contains all but finitely many Hawaiian earrings.
  • If \mathbb{D} is the dyadic arc-space pictured below as the union of the base-arc B=[0,1]\times \{0\} and countably many semi-circles, then \mathbf{w}(\mathbb{D})=B is an interval and thus has uncountably many points.

    dyadircarcspace

    The dyadic arc space

  • It is possible that \mathbf{w}(X)=X! Examples include the Menger Curve, Sierpinski Triangle, and Sierpinski Carpet.

In all of the above examples, \mathbf{w}(X) is closed in X; this is not a coincidence. Proving it’s true in general is a nice exercise to include in a course on fundamental groups.

Exercise: Prove that if X is locally path-connected, then \mathbf{w}(X) is closed in X.

In fact, even in the non-locally path-connected case, something can be said.

More general exercise: Show that if f:Y\to X is a map from a locally path-connected space Y, then f^{-1}(\mathbf{w}(X)) is closed in Y.

An inquisitive reader may wonder in what ways the space \mathbf{w}(X) is a kind of homotopy invariant of X. Some partial answers are published, but I promised myself not to go down the rabbit hole of writing about this yet.

What is algebraic wildness?

In particular, does \mathbf{w}(X)\neq\emptyset, imply that the fundamental group \pi_1(X,x) is wild?

At first glance, it seems like it might. Surely, arbitrarily small non-trivial loops are going to result in complicated algebra, right? To find out, let’s clarify what kind of property “wildness” should be for \pi_1.

  • It can’t be a purely group theoretic property since, by CW-approximation, every group is the fundamental group of some 2-dimensional CW-complex. I don’t care how complicated your group is or how hard the Whitehead conjecture is, the fundamental group of a CW-complex does not count as being wild.
  • An algebraically “wild” fundamental group should admit at least one (possibly trivial) element that is represented by an infinite concatenation of non-contractible loops.
  • It doesn’t just have to happen at the basepoint. Because of path-conjugation, wildness might occur at any point of X.

Therefore, since algebraic wildness is really about the natural presence of infinitary operations like infinite sums in calculus/analysis (and in contrast with binary, trinary, or other finitary operations that come for free in ordinary groups), we’ll use the more descriptive word “infinitary” instead of “wild.”

Definition: A fundamental group \pi_1(X,x_0) is infinitary at x\in X if there exists a loop \alpha:[0,1]\to X based at x, and a closed set \{0,1\}\subseteq C\subseteq \alpha^{-1}(x) with such that [0,1]\backslash C has infinitely many components, and for each component (a_n,b_n) of [0,1]\backslash C, the loop \alpha_n=\alpha|_{[a_n,b_n]} is not null-homotopic. If \pi_1(X,x_0) is not infinitary at x, then we say it is finitary at x. If \pi_1(X,x_0) is finitary at all points of x, we say the group is finitary. If there is at least one point, where \pi_1(X,x_0) is infinitary, we say the group is infinitary.

wildness3

Two possible examples of path decompositions giving rise to an infinite product. The first example realizes an ordinary infinite product and the second case is a dense (or transfinite) product where C is the middle third Cantor set.

Equivalent but more practical definition: A fundamental group \pi_1(X,x_0) is infinitary at x\in X if and only if there exists a map f:(\mathbb{H},b_0)\to (X,x) from the Hawaiian earring such that f restricted to each circle of \mathbb{H} is not a null-homotopic loop.

wildness

Proof of Equivalence: If you start with the first definition, the loop \alpha:[0,1]\to X satisfies \alpha(C)=x and thus induces a map f:[0,1]/C\to X. However, because [0,1]\backslash C has a countably infinite number of components, [0,1]/C\cong \mathbb{H}. Moreover, each loop \alpha|_{[a,b]} induces the restriction of f to a unique circle of \mathbb{H}. Conversely, if we start with a map f:\mathbb{H}\to X, let x be the image of the wild point. If we take \alpha_n to be the loop which is the image of f on the n-th circle, consider the infinite concatenation \alpha=\prod_{n=1}^{\infty}\alpha_n defined as \alpha_n on \left[\frac{n-1}{n},\frac{n}{n+1}\right], and \alpha(1)=x. The set C=\{0,1/2,2/3,3/4,\dots,1\} now satisfies the first definition. \square

Definition: The algebraic 1-wild set of a space X is the subspace

\mathbf{aw}(X)=\{x\in X\mid \pi_1(X,x_0)\text{ is infinitary at }x\}.

Exercise: Construct a locally path-connected space such that \mathbf{aw}(X) is not sequentially closed.

Of course, topological 1-wildness (arbitrarily small non-contractible loops) and algebraic 1-wildness (a shrinking sequence of non-contractible loops) should feel closely related – they both involve the existence of arbitrarily small non-contractible loops at some point.

A lot of the time, they’re the same

The next theorem shows where the two kinds of wildness agree.

Theorem: If X is semilocally simply connected at x, then \pi_1(X,x_0) is finitary at x. The converse is true if X is first countable at x.

Proof. Suppose X is semilocally simply connected at x and f:(\mathbb{H},b_0)\to (X,x) is a map. We must show that f applied to at least one circle of \mathbb{H} is null-homotopic. Find an open neighborhood U of x such that every loop in U based at x is null-homotopic in X. Write \mathbb{H}=\bigcup_{n\geq 1}C_n as the usual union of circles C_n of radius 1/n. By the continuity of f, there is a neighborhood V of b_0 such that f(V)\subseteq U. Given the topology of \mathbb{H}, we may find an N such that C_n\subseteq V for all n\geq N. Therefore, f maps C_N to a loop in U, which must be null-homotopic in X. This completes the first direction.

For the converse, suppose X is first countable at x and that X is not semilocally simply connected at x. Let U_1\supseteq U_2\supseteq U_3\supseteq \cdots be a neighborhood base at x. By assumption, for each n\geq 1, there is a loop \alpha_n:[0,1]\to U_n based at x, which is not null-homotopic in X.wildness2

Define a function f:\mathbb{H}\to X to be \alpha_n on C_n. To check the continuity of f, we really only need to check the continuity of f at b_0: If U is a neighborhood of x, find N with U_N\subseteq U. Then it is clear that f\left(\bigcup_{n\geq N}C_n\right)\subseteq U_N\subseteq U. For n\in \{1,2,\dots N-1\} find a neighborhood V_n of b_0 in C_n such that \alpha_n(V_n)\subseteq U. Now V=\bigcup_{n\geq N}C_n\cup\bigcup_{1\leq n\leq N-1}V_n is a neighborhood of b_0 such that f(V)\subseteq U. Since f is continuous an non-trivial on each circle, \pi_1(X,x_0) is infinitary. \square

Corollary: For any space X, we have \mathbf{aw}(X)\subseteq \mathbf{w}(X) with equality if X is first countable.

So, for first countable spaces, the two notions of wildness are the same. Of course, even with some non-first countable spaces like very large CW-complexes, we still have equivalence.

This all suggests that to find a difference between topological and algebraic wildness, we need to dig into difference between nets and sequences of loops.

but not always

General Topology permits all sorts of phenomenon, including interesting spaces that are not first countable. To conclude this post, I’m going to describe a space that isn’t semilocally simply connected (and doesn’t have a universal covering space) but whose fundamental group is not wild, i.e. is finitary.

Let \omega_1 be the first uncountable ordinal and \omega_1+1=\omega_1\cup\{\omega_1\} be it’s successor, the first compact uncountable ordinal. Here, \omega_1 denote the maximal point of \omega_1+1 and we take it to be the basepoint. The key topological fact we’ll need to remember is that there does not exist any sequence in [0,\omega_1) that converges to \omega_1. Let

X=\Sigma (\omega_1+1)=\frac{(\omega_1+1)\times [0,1]}{(\omega_1+1)\times \{0,1\}\cup\{\omega_1\}\times[0,1]}

be the reduced suspension with canonical basepoint x_0. For each countable ordinal \alpha<\omega_1, the image of \{\alpha\}\times [0,1] will result in a unique circle X_{\alpha} in X. These circles will all be joined at x_0. However, the topology of this space is not the one you’d give to a wedge of CW-complexes. To help visualize this monster, consider the first convergent sequence \{n\}\to\omega in \omega_1+1. This corresponds to the subspace \Sigma(\{1,2,\dots,\omega\}\cup\{\omega_1\}) of X illustrated below as the sequence of circles \{X_n\} converging to the limit circle X_{\omega}.

hoopspace

From this subspace, imagine building X inductively by creating larger and larger bouquet’s of circles parameterized by countable ordinals. In the limit, the circles will “converge” to x_0 in the sense that if U is an open neighborhood of x_0, then there exists a \beta<\omega_1 such that \bigcup_{\alpha\geq \beta}X_{\alpha}\subseteq U.

Sorry, Not Sorry: The space X is compact and it’s locally path connected at x_0, but it’s not locally path-connected at all of its points. I take no responsibility for this. Countable limit ordinals like \omega are to blame. Fortunately, we’re interested in the topology around x_0 so there is no harm in sweeping this under the rug. If you’re unhappy about it, remember that you can take the locally path-connected coreflection without any loss of homotopy/homology group data. The downside to doing so is that the coreflection is not compact. So it goes.

Observation 1: X is not semilocally simply connected. In particular, \mathbf{w}(X)=\{x_0\}.

Proof. Since X is a quotient space of (\omega_1+1)\times[0,1] and [0,1] is compact, for every neighborhood U of x_0, there is a countable ordinal \beta such that \bigcup_{\alpha\geq \beta}X_{\alpha}\subseteq U. Also, X retracts onto each circle X_{\alpha}. Therefore, the loop traversing X_{\beta} is contained in U and is not contractible in X. Since ordinals are totally path disconnected, every path component of X\backslash \{x_0\}\cong \omega_1\times (0,1) is an open interval. Therefore, X is semilocally simply connected at every point in X\backslash \{x_0\}. \square

Observation 2: \pi_1(X,x_0) is finitary, i.e. \mathbf{aw}(X)=\emptyset.

Proof. Since \mathbf{aw}(X)\subseteq \mathbf{w}(X)=\{x_0\}, if \mathbf{aw}(X) contains a point, it must be x_0. Consider any map f:(\mathbb{H},b_0)\to (X,x_0) and suppose, to obtain a contradiction, that f|_{C_n} is not null-homotopic for each circle C_n, n\geq 1 of \mathbb{H}. In order for this to happen, it must be the case that, for each n, there exists some arc of the circle C_n that maps onto some circle X_{\alpha_n}.

In particular, we may find t_n\in C_n\backslash \{b_0\} such that f(t_n) is the image of the point (\alpha_n,1/2) in the circle X_{\alpha_n} (this is the point on X_{\alpha_n} furthest from/antipodal to b_0). Since \{t_n\}\to b_0 in \mathbb{H}, the continuity of f tells us that the sequence \{ (\alpha_n,1/2)\} converges to x_0 in X.

Maybe you already see the problem. Remember the one thing I said we’d need to use about \omega_1+1? There is no sequence of countable ordinals converging to \omega_1. But if (\beta,\omega_1]=\{\alpha\mid \beta<\alpha\leq \omega_1\} is an arbitrary neighborhood of \omega_1 in \omega_1+1, then we may take V to be the open neighborhood of x_0, which is the image of (\beta,\omega_1]\times [0,1]\cup (\omega_1+1)\times [0,1/3)\cup(2/3,1] in X. The only way for the sequence \{ (\alpha_n,1/2)\} to eventually be in V is for the sequence \{\alpha_n\} of countable ordinals to eventually be inside (\beta,\omega_1]. This implies \{\alpha_n\}\to \omega_1; a contradiction of the topology of \omega_1. Since x_0 is not in \mathbf{aw}(X), this set must be empty. \square

Having made Observations 1 and 2, we conclude that even though X fails to have a simply connected covering space, it’s fundamental group is completely tame. In fact, from a fundamental group perspective, it’s basically the same as an ordinary wedge of circles \bigvee_{\alpha\in\omega_1}X_{\alpha}, which is a CW-complex! Intuitively, the neighborhood base at x_0 is so large (more precisely, the “tightness” is so large) that shrinking sequences of loops don’t actually converge to the constant loop at x_0 and this sequential convergence is exactly what is required to have an infinitary fundamental group.

If you put together the ideas behind the proofs of Observations 1 and 2, it’s not too hard to prove the following more detailed results:

Theorem: The continuous identity function \bigvee_{\alpha\in\omega_1}X_{\alpha}\to X from the ordinary (i.e. CW) wedge of circles to X is a weak homotopy equivalence.

Corollary: \pi_1(X,x_0) is canonically isomorphic to the free group F(\omega_1) on uncountably many generators.

Here’s a fun exercise I’ll leave you with.

Challenge Exercise: Construct a locally path-connected space X for which \mathbf{aw}(X) is not closed.

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