Hawaiʻian Earrings
Over on her (very new) weblog, Sarah ponders the Hawaiʻian earring space and the infinite bouquet of circles. She illustrates the earring well, but just to be clear about the latter…
Take one circle with a marked point for each natural number and quotient by an equivalence relation declaring that all those marked points are really the same point. And don’t try to think of it as taking place in any finite-dimensional space. The bouquet of circles has exactly the quotient topology, and not the topology induced by embedding into any finite-dimensional space (that I know of, at least).
Anyhow, a special case of this caveat was exactly what struck her — that the underlying point sets of the earring and the bouquet are clearly isomorphic, but that the topologies are not. In particular, the earring is compact, while the bouquet is not. I believe I’ve discussed enough topology for these two facts to be an exercise for anyone who’s followed everything I’ve said here.
In a comment there, I wave my hands towards one example of where this difference shows up, and I’ll make you go give her some page-views to read it. Show the love!
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The Unapologetic Mathematician 
Isn’t the infinite bouquet just the quotient space R/Z? Or am I missing something obvious?
As I’ve heard the term used it’s the wedge of an infinite number of points. Is that homeomorphic to the quotient space you’re talking about?
Cannot we see the infinite bouquet of circles as the subspace of R^2 formed by the union of the circles centered at (n,0) and of radius n?
That might be the case.. I would have to see an explicit proof first, though.
You’re right to request a proof… because I think this is actually false: the “origin” of the bouquet has more neighborhoods than the origin in my example. For instance, take the image under the quotient map of the union (one in each circle) of open arcs of length $e^{-n}$ around the origin; it is a neighborhood of the origin in the bouquet, but not in the subspace of R^2. (To be precise, whatever the precise function from the bouquet to the subspace of R^2, you can always find such an example (with a sequence decreasing rapidly enough) to prove it’s not a homeomorphism).
I think the answer is no, because in that subspace, you can pick a sequence x_n != (0,0), x_n in the n-th circle, which has (0,0) as its limit. You can’t have this for the infinite bouquet: in the restriction to the n-th circle, pick a neighborhood U_n of the basepoint p which excludes x_n; then the union of the U_n is open in the bouquet.
To be more rough about it (for less topologically-minded people) the circles passing through the base point of the bouquet are “spread out”, and the act of forcing them into any Euclidean space makes them bunch up to an extent that breaks the topology.
I made a comment on Sarah’s site explaining exactly what the CW topology was and why it was strictly finer than the earring topology. (As I mentioned there, since the CW-topology is a textbook example of a “final” or “strong” topology, the fact that the “W” stands for “weak” is especially unhelpful. Better to speak of cell complexes, maybe…)
One more comment, an exercise from a long ago topology course: to give a one-dimensional CW complex is to give a graph. Moreover a 1-d complex is compact iff it has finitely many cells and is locally compact iff only finitely many one-cells get attached to any given 0-cell, i.e., iff the degree of each vertex is finite. Because of this, an infinite bouquet is not locally compact and therefore not homeomorphic to any locally closed subset of Euclidean space.
An interesting question is: which of these topologies is the “correct” one? Answering this may separate the algebraic topologists from the point-set topologists…
Answering this may separate the algebraic topologists from the point-set topologists
Indeed it would, as the former would realize that the bouquet is the real thing, and the earring is the Bonsai Kitten you get by trying to squeeze it into the plane.
I think you are identifying yourself as an algebraic topologist!
A point-set topologist would point out that any bouquet of circles is a rather trivial space: for instance, the computation of all of its homotopy/co/homology groups is an exercise for a student taking a first course.
On the other hand, the Hawaiian earring is much more interesting: people have not “identified” its fundamental group in any satisfactory way, and papers on this space and its fundamental group continue to be written: e.g.
MR2126731 (2006e:20062) Conner, G.; Spencer, K. Anomalous behavior of the Hawaiian earring group. J. Group Theory 8 (2005), no. 2, 223–227. (Reviewer: Andreas Zastrow) 20F34 (55Q05)
MR2280915 (2008b:55017) Fabel, Paul A retraction theorem for topological fundamental groups with application to the Hawaiian earring. Topology Appl. 154 (2007), no. 3, 722–724. (Reviewer: Ross Geoghegan) 55P55 (20E18 57N25)
Biss, Daniel K.(1-MIT)
A generalized approach to the fundamental group.
Amer. Math. Monthly 107 (2000), no. 8, 711–720.
In particular, the Hawaiian earring is an example of a space whose “topological fundamental group” is not discrete.
I didn’t say that the earring isn’t more complicated. In fact, the simplicity of the bouquet is what identifies it as the natural object.
But you’re right, that’s exactly what I’m doing.
The modifier “correct” in the question at the end of comment 8 strikes me as an odd choice. My first inclination is to interpret it just as John did: what is the “most natural” choice of topology? And since the underlying set is an appropriate union (colimit) of circles, the answer would seem to be the colimit topology, which gives the bouquet.
Would the Rorschach-like intent of the question be better conveyed if “correct” were replaced by something like “more important”? From one point of view, the bouquet probably gets “used more”, being a universal construction (e.g., consider how CW complexes get built up, as in Brown’s representability theorem). But the Hawaiian earring is arguably more interesting, at least for many temperaments; it’s certainly arguably more pathological.
Something like this poll could be conducted for product topology vs. box topology, although I’m not sure how best to formulate the question so as not to bias the results (I’d consider the product topology vastly “more important”).
You’re absolutely right that the word “correct” is, in and of itself, meaningless. It was an attempt to hide the subjectivity inherent in a question like “Which one do you prefer?”
It should at least be possible to argue that the earring is less pathological than the bouquet: after all, despite what was temporarily claimed above, only the Hawaiian earring is realizable as a compact subspace of the Euclidean plane. And if you have two Hausdorff topologies on the same set, one of which is finer than the other, and one of which is compact, which one seems more natural?
I think the two sides are relatively evenly matched (and, indeed, I am neither a point-set nor an algebraic topologist).
On the other hand I cringe with revulsion at the thought of someone who would prefer — for any reason — the box topology to the product topology. What does that say about me?
I don’t know, except that we can’t conclude from this test that you’re nuts?! 🙂 Seriously, I think there are probably many people, many point-set topologists in particular, who delight in weird pathologies, and the box topology seems to be a good source of ’em. But I’m with you — that just ain’t my style.
I can sort of see your point about compact Hausdorff topologies, except that there are jillions of ways of putting a compact Hausdorff topology on a countable join of pointed circles (in such a way that you get the usual topology when you restrict to each circle), and I have a hard time choosing one among them that stands out as the “natural” one. [I don’t call myself a topologist of whatever stripe either, but to the category theorist in me, the universal topology stands out! :-)]
And really what else is an algebraic topologist but an applied category theorist?
Off the top of my head, I would have guessed that the earring tpology is the only compact metrizable (equivalently, second countable) topology on the set in question which restricts to the usual topology on each circle. Having thought about it more, I may need to add the following condition, which however seems very “reasonable” (and is also satisfied for any finer topology, like the CW topology):
Each nonwedge point Q on a given circle C_n has an open neighborhood in the entire space which is contained in C_n.
Indeed, suppose the distinguished point P had a neighborhood U such that there are infinitely many integers n_k such that the intersection of U with the n_k-th circle is proper. For each k, a point x_k in C_{n_k} \setminus U. Then {x_k} does not have any subsequence converging to P. But it certainly can’t have a subsequence converging to any other point, by the condition imposed in the last paragraph above. Thus the space is not sequentially compact and therefore not compact metrizable.
Again, I think the earring space (unlike the box topology) has very nice behavior with respect to all of the properties that are studied in point-set topology. What it isn’t is semilocally simply connected, but that’s a property that seems basic in algebraic topology, not point-set topology.
“And really what else is an algebraic topologist but an applied category theorist?”
In Glory Road, a fantasy novel by Robert A. Heinlein, serialized in The Magazine of Fantasy & Science Fiction [July – September 1963] and published in hardcover later the same year, the protagonist is told (in an alternate world where magic works), that magic is merely “applied topology.”
After saving the multiverse, said protagonist, E.C. “Easy” Gordon, recently discharged from an unnamed war in Southeast Asia, takes courses at Caltech.
Pete, I think your conditions as amended probably do uniquely characterize the earring (and without recourse to embeddings in Euclidean space), and it that in some sense the two sides do seem somewhat more “evenly matched” to me now. Still, rightly or wrongly, I can’t help but think that the bouquet is more “fundamental”, in that it gets used far more often in constructions, whereas the earring strikes me as an interesting curiosity, studied more or less for its own sake, or as an exceptional example to test things on. But I’ll admit I don’t know much about it.
Re “pathology”: you’re right, “box topology” is not a terribly apt analogy. “Pathology” is a slightly loaded term anyway; I had in mind more of a “Counterexamples in Topology” meaning (cf. the book with that title by Steen and Seebach, which contains lots of examples of compact Hausdorff subspaces of R^n as, um, counterexamples).
Speculating out loud, there might be some logical sense in which the earring is “wild” whereas the bouquet is “tame”. Both spaces are particularly simple models for the classifying spaces of their fundamental groups; an idle conjecture is that the fundamental group of the earring is undecidable (has an undecidable theory in the language of groups) whereas the fundamental group of the bouquet is not. And that this phenomenon might carry over to the logical theories of these spaces as well, without particularly needing to stack the deck with the algebraic topologist’s term “classifying space”. But I’m going out on a limb here; haven’t thought about this too carefully.
I agree with you — the earring is not really pathological and is, considered purely as a topological entity in its own right, arguably more interesting than the bouquet of circles. On the other hand, the bouquet of circles — or rather, any bouquet of circles, finite, countable or otherwise — is indubitably more useful. For instance (as I’m sure you know), one can use such bouquets to give a very nice proof that a subgroup of a free group is free. I have never seen an important result in which the Hawaiian earring appeared in the proof but not in the statement….
Anyway, it was an interesting discussion.
One cool fact about the Hawaiian earring is that it is homeomorphic to the 1 point compactification of a countable collection of open intervals (equivalently real lines). Furthermore, Brin and Thickstun (I believe) point out that the universal cover of a non-trivial knot exterior is homeomorphic to a 3-ball with a Hawaiian earring removed from its boundary. So I consider the earring to be the more natural 🙂