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Is there a standard construction of a metric on one-point compactification of a proper metric space?

Comments:

  • A metric space is proper if all bounded closed sets are compact.
  • Standard means found in literature.

From the answers and comments:

Here is a simplification of the construction given here1 (thanks to Jonas for ref). Let $d$ be the original metric. Fix a point $p$ and set $h(x)=1/(1+d(p,x))$. Then take the metric $$\hat d(x,y)=\min\{d(x,y),\,h(x)+h(y)\},\ \ \ \ \hat d(\infty,x)=h(x).$$

A more complicated construction is given here2 (thanks to LK for ref), some call it "sphericalization". One takes $$\bar d(x,y)=d(x,y)\cdot h(x)\cdot h(y),\ \ \ \ \bar d(\infty,x)=h(x).$$ The function $\bar d$ does not satisfies triangle inequality, but one can show that there is a metric $\rho$ such that $\tfrac14\cdot \bar d\le \rho\le \bar d$.

1Mandelkern, Mark. “Metrization of the One-Point Compactification.” Proceedings of the American Mathematical Society, vol. 107, no. 4, American Mathematical Society, 1989, pp. 1111–15, https://doi.org/10.2307/2047675.

2Mario Bonk, Bruce Kleiner: Rigidity for Quasi-Mobius group actions

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  • $\begingroup$ Is there a standard strictly increasing function from $\mathbb R\ge0$ to $[0,1)$? If you make your metric finite in this way, I should think you would get a metric on the compactification by adding in the 1 for a distance to the point at infinity. Not an answer, because I may be wrong. $\endgroup$
    – Elizabeth S. Q. Goodman
    Commented Mar 18, 2010 at 3:19
  • $\begingroup$ What do you mean by "standard": found in literature or canonical w.r.t. some class of maps? $\endgroup$
    – Sergei Ivanov
    Commented Mar 18, 2010 at 3:40
  • $\begingroup$ @Sergei. "standard" = "found in literature" $\endgroup$
    – Anton Petrunin
    Commented Mar 18, 2010 at 3:50
  • $\begingroup$ @Elizabeth, you are right, that very much like your question, but I need to work with particular choice of metric and if there is one people already use I would be happy to use the same (especially if it already has a name). $\endgroup$
    – Anton Petrunin
    Commented Mar 18, 2010 at 4:01
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    $\begingroup$ I'm guessing you've considered this because it is one of the first things that came up in a Google search, but how about section 3 of jstor.org/stable/2047675? I don't know if it is relevant because they use the term "one point compactification" in an unusual way, but perhaps for proper spaces they are the same? $\endgroup$
    – Jonas Meyer
    Commented Mar 18, 2010 at 4:10

2 Answers 2

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I will make this an answer although it is just a follow-up to the comment of LK

The recent names in this (but referring back to Bonk and Kleiner) are Stephen Buckley and David Herron, for proper spaces their one-point extension $\hat{X}$ is the one-point compactification, see pages 4 and 8 in

[PDF] METRIC SPACE INVERSIONS, QUASIHYPERBOLIC DISTANCE, AND UNIFORM SPACES File Format: PDF/Adobe Acrobat - Quick View by SM Buckley - 2008 - Cited by 3 - Related articles In a certain sense, inversion is dual to sphericalization. ... point compactification. All of the properties of dp mentioned above also hold for their con- ...

https://eprints.nuim.ie/1610/1/BuckleyMetricSpace.pdf

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There is a standard metric for the one-point compactification of the complex plane, given by stereographic projection of the sphere onto the plane. But of course if you translate the plane first, the result is different, so is even that an example of what you want?

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    $\begingroup$ Do you see a simple way to generalize it to proper spaces? $\endgroup$
    – Anton Petrunin
    Commented Mar 18, 2010 at 17:12

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