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stereographic projection gives a homeomorphism between $S^1$ and the real number line with an added point at infinity, so that whether you journey off in the distance to negative infinity or positive infinity, you're heading towards the same point. Even more wonky to me is $S^2$ to the plane with a point at infinity...! I understand these one point compactifications are topologically convenient, and that also having a single infinity makes a lot of sense when the plane has a complex structure, and yet something just seems philosophically/intuitively wrong to me about this!!

Can somebody give me some more reasons why this should not bother me? That no matter what direction you go, you are heading towards the same infinity? Obviously negative infinity and positive infinity are not meant to always be identified as the same thing...

RobPratt
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PhysicsIsHard
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    When pondering "infinity" from a mathematical standpoint, it is a good idea to sweep your brain clean of the age-old philosophical conundrums of "infinity". On this problem, just sit yourself down and force yourself to really learn about the one-point compactification of a topological space, and how it applies to the real number line. – Lee Mosher Jul 30 '25 at 23:27
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    "Obviously negative infinity and positive infinity are not meant to always be identified as the same thing..." right, and stereographic projection is not a tool to be applied in every situation. There is a compactification of $\mathbb R^2$ which adds a circle at infinity, and that's more useful in some contexts. – Cheerful Parsnip Jul 31 '25 at 02:35
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    There's also the Stone Cech compactification, which adds unfathomably many different points at infinity. In fact it adds all the infinities you will ever need. – Trebor Jul 31 '25 at 08:36
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    Complaining about a construction some people uses is kind of equivalent to complaining that some people use the axiom of choice in proofs without thinking about it (perhaps even without being aware). If you don't feel that's right, ignore those people and do what seems right to you (while letting them do what seems right to them). – Henrik supports the community Jul 31 '25 at 09:00

3 Answers3

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Can somebody give me some more reasons why this should not bother me?

The fact that you are bothered means you are in good company. Historically, whenever the notion of "number" has had its meaning extended, there were people who were "bothered" by the extension and thought that "something just seems philosophically/intuitively wrong". Examples of this include:

  • the idea that zero is a number

  • the idea that numbers can be negative

  • the idea that numbers can be non-integer

  • the idea that numbers can be irrational (major crisis there!)

  • the idea that numbers can be non-real

  • the idea of infinity (and particularly the idea that there is more than one type of infinity).

What we can learn from this history is that, first of all, it is natural and common to feel discomfort when learning about new concepts that extend or generalize the meaning of concepts we were already familiar with (particularly related to the notion of "number"); and second of all, that that discomfort has not so much to do with the nature of the mathematical concepts that cause the discomfort, but more to do with the psychological attachment that humans tend to form with notions we learn and get used to. We simply tend to resist the idea that a word or concept that we have gotten used to interpreting in a certain way has now broadened its meaning.

If you got used to thinking of "infinity" as meaning one particular thing and find that the one-point compactification notion of infinity pushes you beyond that relatively narrow idea and causes you discomfort, think of that discomfort as being a tangible sign that your brain is expanding and rewiring itself. I find it a bit analogous to the pain (which many people find pleasurable, at least at moderate levels) that you get when you do physical exercise - people enjoy that feeling because they see it as a sign that their muscles are growing and that they're becoming more fit. In the analogous math context, the discomfort can be a small reminder that you are becoming smarter. (Conversely, if you are reading a math textbook and not feeling any discomfort, there's a decent chance you're not actually thinking hard enough about what you're reading to really digest the ideas being discussed.)

In general, when one learns higher mathematics, there is a point at which a student stops constantly asking "but why does this definition make sense" and learns to accept that the explanation of why a concept is interesting or useful often comes after (sometimes long after!) the definition has been formulated. In the interim period between learning the definition and getting the satisfaction of understanding why it's a good one, one often experiences a tension or mild discomfort. That's okay! And some people not only get used to this feeling, but even learn to savor it and enjoy the anticipation of the moment when everything will become clear.

Of course, in this particular case, you do not need to wait very long to understand why having only one infinity for different directions can be useful - some of the other answers discuss this and have given pretty good explanations. But my analysis above applies more generally to many situations where you will learn an abstract mathematical concept and it won't necessarily make sense right away or will appear to be at odds with some earlier knowledge you had. So I hope you find this point of view helpful.

Dan Romik
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    Or just learn to accept that "Why is this definition useful?" is sometimes a better question than "Why does this definition make sense?" – Stef Jul 31 '25 at 11:34
  • @Stef I’m not sure I understand the distinction. Can you give an example of a definition that is useful but doesn’t make sense, even after you understand why it’s useful? – Dan Romik Jul 31 '25 at 13:19
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    The OP cannot "make sense" of the one-point compactification of the plane because it defines a single "point at infinity", which clashes with the notion of "infinity" they already have. Asking "How does this definition make sense?" doesn't really have an answer at all. It's just a definition. It presupposes that the word "infinity" is free and uses it without caring about other possible definitions tied to that word in other contexts. But it's a useful definition, because we can then use it to do cool things with stereographic projection. – Stef Jul 31 '25 at 14:42
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    @Stef: I don't agree with you that the question "How does this definition make sense" has no answer. The sense that one makes out of a new mathematical definition builds and builds, as one learns more and more about it: as one learns theorems about those newly defined mathematical beasts; as one reads new definitions that build upon that old one; and so on and so on... – Lee Mosher Jul 31 '25 at 15:04
  • @Stef thanks for clarifying. I think there can be cases where a definition is useful but doesn’t make sense because the words it uses don’t correspond to the notion it’s trying to capture. E.g., if instead of “the point at infinity” we called it “the stinky point”, the term wouldn’t make sense even after you understood why it was useful. But “the point at infinity” does make (a lot of) sense after you understand why it’s useful. So I agree with Lee, in this situation the usefulness, together with the words chosen for the definition, do build up to the concept (eventually) “making sense”. – Dan Romik Jul 31 '25 at 21:42
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Different compactifications have different uses.

To build the projective plane from the plane you add a point at infinity for each direction (pencil of parallel lines). That leads to some nice geometry.

The one point compactication to the 2-sphere is more than topologically convenient. Stereographic projection is conformal: angles are preserved. So much of the geometry in the plane translates to a version on the sphere.

Ethan Bolker
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  • I think it also worth notice that both compactifications are very useful not only for Geometry but also for Topology and Algebra. – jjagmath Jul 30 '25 at 21:27
  • Yup, people can add as many ideal points as they find useful. I think it's worth mentioning that the protective plane isn't the ultimate answer - think about an infinite spiral, like $r =\theta$. It doesn't approach any particular point in the protective plane, but it does in the one point compactification. – JonathanZ Jul 30 '25 at 21:37
  • @jjagmath Is the one-point compactification very useful in algebra? (Except when it agrees with some projective space, i.e. $S^1$, $S^2$ and maybe $S^4$, $S^8$?) – ronno Jul 31 '25 at 11:38
  • @ronno Ethan's answer is referring to the one-point compactification of the complex plane, not the one-point compactification of topological spaces in general. – jjagmath Jul 31 '25 at 12:10
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It is useful to keep in mind that one is free to embed a space in another space in whatever way is useful. For instance the one-point compactifications of the $S^n$ make them homogeneous -- each pair of points are topologically equivalent.

However, you might want to read about topological ends, which is a compactification technique that captures the different directions. You might also look at blow-ups (oversimplifying: ) a way to un-one-point-compactify (deleting a singularity) and recover the various paths approaching the boundary. See also crosscap, on which a point has been extruded into a line segment and each angle through that point has been separated to a different point on that line segment. (The line segment is double-covered in the projection to 3-space.)

Eric Towers
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