# Metric on one-point compactification

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

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

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

• 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. Mar 18, 2010 at 3:19
• What do you mean by "standard": found in literature or canonical w.r.t. some class of maps? Mar 18, 2010 at 3:40
• @Sergei. "standard" = "found in literature" Mar 18, 2010 at 3:50
• @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). Mar 18, 2010 at 4:01
• 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? Mar 18, 2010 at 4:10

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