No that is not true. You would need to define your infinity. Here we can see you mean the one point compactification of the real line. There is a two point compactification where they are different.
That is not how one defines the one point compactification topologically.
In what sense are you using base? Zero isn't a base in the usual mathematical sense for numbers.
?????? You want some new metric on the real numbers using the standard metric from the circle.
As we are using the one point compactification, surely there are two numbers halfway between 0 and infinity?