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I have seen this question about generalising divergence to infinity in a direction, but I believe the discussion is unclear as one would have to describe what to describe what diverging to infinity means in general even means.

A comment by Sangchul Lee on a previous post of mine gives a hint,

The notion of divergence to ±∞ requires more than merely metric space structure. Recall that this notion is equivalent to convergence to +∞ or to −∞ in the extended real number line. Analogously, to define divergence to a point of infinity on any metric space M , we need a choice of compactification of M

With this hint, my question becomes, why is compactification of a metric space the required to generalize divergence to infinity?

tryst with freedom
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    It is a mistake to assume that a compactification is needed. For instance, in some cases, Gromov boundary will not be compact (if the original space is not proper). – Moishe Kohan Apr 17 '23 at 17:47
  • Related – Mittens Apr 17 '23 at 18:06
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    It can often be simpler to think of limits as a more general concept relating filters on the domain and target spaces. In the case of a nonempty metric space $X$, you can think of fixing an element $x_0$, and then forming the filter generated by ${ X \setminus B_R(x_0) \mid R > 0 }$. It turns out that that filter doesn't depend on the choice of $x_0$. If you'd like to see an answer explaining that in more detail, then I can try to write one. – Daniel Schepler Apr 17 '23 at 18:22
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    I also disagree that the discussion in the linked question is unclear: You just have to read more, starting with the suggested reference, to understand different ways of adding "points at infinity." Another useful reference is the book by Burago, Burago and Ivanov "A course in metric geometry", pages 168-172. – Moishe Kohan Apr 17 '23 at 18:23
  • I am not immediately sure of the utility of this idea of filters, but would love to see some discussion on it still. @DanielSchepler Thanks – tryst with freedom Apr 17 '23 at 19:17
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    This is your third recent question (one you have deleted) asking for a version of this kind of generalization. It's unclear (to me) just what you want - and it may not exist. I suggest you follow up on many of the suggestions in comments and answers. – Ethan Bolker Apr 17 '23 at 21:16
  • The first question I asked was quite a mess. I realized I had some basic misunderstandings, and after some more analysis, found it was split into the two new questions I have posted. I do not plan on undeleting the first deleted question. As far as incoporable things go, if you check the other post, you would see that I have gone through almost all posts and taken their suggestion except that of Dave L. Theirs, I will require some more time, as it is the first time I heard of the concept they mentioned. For this q, I am not sure, which points can be incoporated. @EthanBolker – tryst with freedom Apr 17 '23 at 21:20

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Maybe not required, but it gives an easy way to make a definition. You don’t want ‘$\infty$’ to be in your space, so you need to adjoin something, and the topology given by compactifications just works in the special cases so it’s a nice way to generalise.

The one new point that is adjoined in the Alexandroff compactification is often referred to as the ‘point at infinity’ and “divergence/convergence to infinity” can be made precise by: the object converges to the ‘point at infinity’ in the Alexandroff compactification.

Indeed, “$\lim_{z\to\infty}$” is often seen in complex analysis and it almost always means (something equivalent to) “the limit as $z\to\omega$ if $\omega$ is the adjoined point in the Alexandroff compactification”.

If you use a two point compactification you could get a generalisation of “$+/-\infty$”.

Compactifications adjoin points that have to be made ‘close to everything’ that are at the same time not already in the space. Picture turning $\Bbb R^2$ into $S^2$ by rolling up the plane and making - as you move out in all directions - it converge to the North Pole.

N.B. I’ve also seen the points adjoined in the Stone-Cech compactification referred to as points at infinity.

FShrike
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