# 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 here (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 (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 $\tfrac14\cdot \bar d\le \rho\le \bar d$.

• 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. – Elizabeth S. Q. Goodman Mar 18 '10 at 3:19
• What do you mean by "standard": found in literature or canonical w.r.t. some class of maps? – Sergei Ivanov Mar 18 '10 at 3:40
• @Sergei. "standard" = "found in literature" – Anton Petrunin Mar 18 '10 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). – Anton Petrunin Mar 18 '10 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? – Jonas Meyer Mar 18 '10 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