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 here^{1} (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 here^{2} (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 $\tfrac14\cdot \bar d\le \rho\le \bar d$.

^{1}Mandelkern, 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.

^{2}Mario Bonk, Bruce Kleiner: *Rigidity for Quasi-Mobius group actions*

4more comments