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1. I have changed the title of this article, as well as references to the object within it. Use of the term “Hawaiian Earring” is objected to by Hawaiian mathematicians. Please see these two threads, one by native Hawaiian and math PhD Dr. Marissa Loving, and the other by an expert on the Hawaiian Earring, Dr. Jeremy Brazas.

I have retitled the article “Shrinking wedge of circles”, which is the name used for this space in Hatcher’s “Algebraic Topology”. I have a retained a note in the body of the article that the space is sometimes referred to as the “Hawaiian earring space”.

This small change in name helps to make mathematics a more inclusive and just field, especially in consideration of the historical marginalization and exclusion of indigenous mathematicians. By taking this action, the nLab site can help to spread a change in language more widely, including on other math reference sites.

I hope that this change is readily accepted and approved by the nLab community. Thank you!

Justin Lanier

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJun 20th 2021

Thank you! I endorse that edit. I added a redirect for Hawaiian earring space.

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 20th 2021
• (edited Jun 20th 2021)

MathSciNet lists over a hundred papers that use the Hawaiian earring terminology.

It lists only 3 papers that talk about shrinking wedges (but not as a designated name for a space, only as an informal description), and all three also use the term Hawaiian earring.

Hatcher’s book appears to be an aberration in this particular aspect.

Also, changing the terminology appearing in hundreds of papers just because one or two mathematicians object to it does not make any sense.

• CommentRowNumber4.
• CommentAuthorHurkyl
• CommentTimeJun 20th 2021
• (edited Jun 20th 2021)

The extremely jingoistic rationale given for the change is concerning.

Regarding the second twitter thread that recounts his own use of “Hawaiian” as a metaphor for “exotic”, is that something people actually do, or was that just his circle? I’ve literally never heard the term “Hawaiian” used in mathematics outside of this particular proper noun. The only other usage I could find after a brief search is to constructions made out of the Hawaiian earring (e.g. “Hawaiian group”).

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeJun 20th 2021

@Dmitri,

at the cost of perhaps being contentious, compare the use of “he” for the generic reader in past mathematics papers. We cannot change the literature, but we can change terminology. Mathematical terminology changes all the time, and this particular topological space is not so common relatively speaking (compare the term “space” itself, which has changed! At least among homotopy theorists). Also, a descriptive name is far better than one named for a place or a person (“profinite set” is more informative than “Stone space”, for instance).

• CommentRowNumber6.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 21st 2021
• (edited Jun 21st 2021)

One of the earliest appearances of the terms “Hawaiian earring” and “clamshell” are in “Continuity of homomorphisms” by R. M. Dudley:

Duke Math. J. 28(4): 587-594 (December 1961). DOI: 10.1215/S0012-7094-61-02859-9

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 21st 2021

not so common relatively speaking

It appears in hundreds of papers. MathSciNet yields over a hundred matches, and it doesn’t do full text search, only titles and reviews.

Also, a descriptive name is far better than one named for a place or a person (“profinite set” is more informative than “Stone space”, for instance).

I disagree, and your example illustrates the problem with your approach perfectly: the equivalence of categories between profinite sets and Stone spaces is nontrivial. How do you even formulate it if you are not allowed to use the term “Stone space”, other than clumsily repeating “compact totally disconnected Hausdorff space” every time you need it?

And then we also use “descriptive” names like regular space and normal space.

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 21st 2021

Re #5:

We cannot change the literature, but we can change terminology.

If the terminology changes in the published literature (i.e., a lot of journal articles and books use some new name), eventually the nLab can be updated accordingly.

Right now, this change is completely absurd: searching MathSciNet using the proposed new name reveals nothing, when in fact hundreds of papers are available on the topic.

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeJun 21st 2021

I don’t propose never mentioning “Hawaiian earring” ever on the nLab, so that people can’t tell what’s going on. If the page title is “shrinking wedge of circles”, and then the intro mentions that the historical name is “Hawaiian earring”, then there is plenty of scope for continuity.

But I’m not going to fight about it, and will let others sort themselves out.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJun 21st 2021

This looks like vandalism by a complete newbie to me. The discussion is appropriate for Twitter and should please be had there.

I was about to revert, but now I see that the situation is worse than I thought: Right now we can’t rename the entry to its original title because the OP also created (here) a new entry with the title “Hawaiian earring space” [sic].

We need to:

• clear and remove that new duplicate entry,

• rename the old entry back to its original title.

Will do so, but first I need to take care of something else.

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeJun 21st 2021

Maybe let’s tone down the language in response. Of course people can disagree with the action(s), but “extremely jingoistic”, “vandalism”, “complete newbie” sound pretty antagonistic.

I’ll go along with #9 otherwise.

• CommentRowNumber12.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 21st 2021

Added this text, and lots of redirects:

The term “Hawaiian earring” is by far the most common name for this space and is used in hundreds of papers. Rarely, other terms can be found in the literature, such as “clamshell”, “infinite earring”, and “shrinking wedge of circles”. \cite{Dudley} is an early reference that mentions the terms “clamshell” and “Hawaiian earring”.

• CommentRowNumber13.
• CommentAuthorDavidRoberts
• CommentTimeJun 24th 2021

Gave earlier reference for this space, where it not given any special name. Dudley cites Griffiths’ paper when giving the definition.

If one is being cheeky, one could call it the “Griffiths space”, assuming this is the first place it appears (of course, it may not be due to Griffiths, but when has that stopped people? :-P)

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJun 24th 2021
• (edited Jun 24th 2021)

I have slightly rewritten and expanded the Idea-section (also added a graphics, grabbed from Wikimedia here), to make it say more about the space itself before discussing terminology.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeJun 24th 2021
• (edited Jun 24th 2021)

By the way, this type of earring is sold as “Italian earrings” here and here.

• CommentRowNumber16.
• CommentAuthorDavidRoberts
• CommentTimeJun 24th 2021

Thanks, I like the intro now. “More recently” is a good way to put it, given that the current page name appears to be staying put.

And the one-point compactification definition is really nice!

• CommentRowNumber17.
• CommentAuthorDavidRoberts
• CommentTimeJun 24th 2021
• (edited Jun 24th 2021)

Notice that up to homeomorphism (which is the best we can expect with the one-point compactification model), we could instead describe the earring space as the one-point compactification $\left(\bigsqcup_{\mathbb{N}} \mathbb{R}\right)^+$. (Edit: or even $(\mathbb{Z}\times \mathbb{R})^+$, which is rather pleasing!)

Following this observation, a nice model that just occurred to me is the pencil of real lines $(Re z=n)_{n\in \mathbb{Z}}$ in $\mathbb{C}\subset \mathbb{CP}^1$ together with the point $\infty\in \mathbb{CP}^1$, at which they all meet. I think this has the right topology!

• CommentRowNumber18.
• CommentAuthorDavidRoberts
• CommentTimeJun 24th 2021

Added description as one-point compactification of $\mathbb{N}\times \mathbb{R}$.

• CommentRowNumber19.
• CommentAuthorDavidRoberts
• CommentTimeJun 24th 2021

Added the fact the earring space is a metric space to the discussion that it is not particularly pathological as a topological space.

• CommentRowNumber20.
• CommentAuthorDavidRoberts
• CommentTimeJun 24th 2021

Added reference for the one-point compactification description.

Also added a description as the suspension of the free convergent sequence, i.e. the one-point compactification of $\mathbb{N}$. I guess $\mathbb{N}$ could be replaced by any countable, infinite discrete space here.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeJun 25th 2021

I have given the discussion of properties a Properties-section. At the end of the first paragraph there I have added the previously missing conclusion that therefore the space is not semi-locally simply connected.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeJun 25th 2021
• (edited Jun 25th 2021)

Added image of the actual earrings (via #15).

• CommentRowNumber23.
• CommentAuthorHurkyl
• CommentTimeJun 25th 2021
• (edited Jun 25th 2021)

Regarding #15, it looks more like it’s a type of “nested circle earring”, which turns up a number of similar search results, with your first example being near the top of my search results. Searching for “italian earring” doesn’t really turn up similar results.

From the product details of your examples, as well as my own searches, it looks like the language convention is that “italian” is in reference to the country where it was made or the nationality of designer. I thought it might have been a reference to the kind of designs and styling of the metal, but I’m less sure on that now. Maybe both usages happen.

I was kind of surprised that a lot of “Hawaiian jewelry” didn’t look distinct from the sort of jewelry I was already vaguely familiar with. I’ve learned from my searches that it looks like native Hawaiian jewelry styles were derived from gifts that Europeans gave to Hawaiian royalty, so it really is the same general kinds of styles. I’m not sure how confident I should be in that fact, though, so take it with a grain of salt.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeJun 25th 2021

Right, I found these by searching for “nested circle earrings”. Still, it’s good to see, for normalization of the discourse, country names attached to the design.

• CommentRowNumber25.
• CommentAuthorHurkyl
• CommentTimeJun 25th 2021
• (edited Jun 25th 2021)

I’m not sure I understand the comment in #24. It sounds like your goal was just to pick something else in the “{national adjective} earring” category which doesn’t make sense given the comment that prompted this discussion nor in the context of the objections that the current name is well-established.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeJun 25th 2021

At monster group there is a picture of a monster, and now at Hawaiian earring space there is a picture of such an earring. Makes sense to me and also makes me chuckle. But it’s not important. If it causes a headache, feel free to remove.

• CommentRowNumber27.
• CommentAuthorHurkyl
• CommentTimeJun 25th 2021

Ah, having a picture of an earring is fine, I was confused because I thought you were proposing “italian earring” as a correct name for the jewelry and as a name for the space.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeJun 25th 2021
• (edited Jun 25th 2021)

I just thought it serves to normalize the level of excitement to see other country names being casually attached to a piece of jewelry, as people would have done in saner times. I don’t expect this to be Italian in any specific way. The design seems rather timeless. Following a whim I checked if I could find it in the Minoan culture or the like. But I have other things to do…

In any case, I cropped the image in the entry a little. The previous version was maybe a bit of an overkill. Now it looks smoother, I suppose. The image is still hyerlinked, though, to QVC selling nested circles for 130 bucks.

• CommentRowNumber29.
• CommentAuthorDavidRoberts
• CommentTimeJun 25th 2021

Reduced suspension, not suspension, and specified the basepoint. This means the space can be conjured out of nothing by universal properties, and even in eg HoTT.

• CommentRowNumber30.
• CommentAuthorDavidRoberts
• CommentTimeJun 25th 2021

To justify that last statement, I think one can start from nat, do the procedure outlined by Martin Escardo in these tweets to get the (compact, non-discrete) type that is the one-point compactification, and then smash with $S^1$. Aside from the the step Martin outlines (which he has done in Agda, of course), even I know how to do all these in HoTT. Thus spaces with ’”exotic” covering space theory/fundamental groups can turn up as the result of combining otherwise nice functors/universal properties (I guess the compactification step is highly nontrivial, and where all the fun is happening, but … shhh)

• CommentRowNumber31.
• CommentAuthormartinescardo
• CommentTimeJun 25th 2021
• (edited Jun 25th 2021)

I have a very minor quibble about something added to this page, namely that it doesn’t matter that the circles get smaller and smaller. It does and it doesn’t. It very much does if you want to implement the earring as a subspace of the Euclidean plane. It doesn’t when you describe the earring topologically, independently of this planar representation, as the one-point compactification of a disjoint sum of open intervals. Although the quibble is minor, perhaps we should avoid potential confusion? You pointed to my tweet sequence, and so now I point to yours in return: https://twitter.com/HigherGeometer/status/1408025411959754754?s=20, in particular regarding John Baez’s comment which led to the current statement in the earring page I am not particularly happy about.

• CommentRowNumber32.
• CommentAuthorR.Arthur
• CommentTimeNov 29th 2021
• (edited Nov 29th 2021)

Remove mobile Wikipedia link and replaced it with the desktop one. When on mobile you get send to the mobile page anyways.