Is $0$ the midpoint of $(-\infty,+\infty)$?
Intuitively, I'd think so, and trying to refine my intuition as to why I'd think so, I would say that this is the case because there is a one-to-one correspondence between $(-\infty,0)$ and $(0,+\infty)$. So there's an equal amount of numbers on either side.
On the other hand, there's a one-to-one correspondence between $(-\infty,k)$ and $(k,+\infty)$ for all $k\in\mathbb R$, which suggests that any real number is a good midpoint. This obviously does not match the intuition for what I mean by "midpoint".
So, the two questions are:
- Is $0$ a midpoint of $(-\infty,+\infty)$?
- If the notion of midpoint is inconclusive or vague when applied to $(-\infty,+\infty)$, is there another notion I could apply which captures my intuition about $0$ being the "midpoint"?