1

I know this has been asked before but I still had some questions after reading through the answers. Anon explains how in a metrizable topology you can decide which of two numbers $a , b$ are closer to infinity by examining connected neighborhoods that contain $+\infty$, $a$ and $b$ and seeing which one would be excluded first when shrinking the neighborhood. For me, this doesn't seem to answer the fundamental question: can we get closer to infinity? If we can make a connected neighborhood $[0,\infty]$, doesn't that assume there is some $ x \in {\rm I\!R}$ that is arbitrarily close to $+\infty$, thus just reasoning of the assumption that we can get close to infinity in the first place?

Maybe I'm wrong in thinking this but if you could find a (short) connected path from $[0,\infty]$ then doesn't that mean you could find the largest value of ${\rm I\!R}$ which is a contradiction? I think there is a good chance there are some fundamentals about Topology that I am missing so I would appreciate anyone chiming in.

P.S. As just kind of an afterthought, I don't know why the idea of approaching $0$ seems so much more intuitive to me when it's really the same concept, inverted. This is probably just a visualizing problem I have with understanding limits in general.

Community
  • 1
Connor James
  • 183
  • 2
    $\mathbb{R}$ is not a point, so what would be the meaning of its distance to any specific other point? – coffeemath Aug 26 '16 at 01:46
  • I suppose it depends which topology you are referring to. If you look at the Riemann Sphere and use the associated topology then one can very easily talk about those points which are arbitrarily close to $\infty$ as it appears on the sphere. Indeed, for any desired distance $\epsilon$, one can find a point within the distance $\epsilon$ to $\infty$ on the sphere. Just because we can always do this however does not imply that there is a "closest" number that we can point to. – JMoravitz Aug 26 '16 at 01:53
  • Thank you for your comments so far, they've helped a lot. I keep thinking of infinity as a value that's so large that you can take away from it and it'll still be infinite but it seems like there are plenty of fields that make valid points about why that is not the case. – Connor James Aug 26 '16 at 02:10

1 Answers1

2

Let me ask a related, but easier, question: consider a short connected path from $0$ to $1$ in $\mathbb{R}$ (I think you mean short connected path from $0$ to $\infty$, rather than from $\mathbb{R}$ to $\infty$?). Does the existence of such a path imply a largest value of $[0, 1)$?

The picture is the same for the extended real line under an appropriate metric, but with $+\infty$ in place of $1$. Topologically speaking, there's no way to distinguish $\mathbb{R}\cup\{+\infty, -\infty\}$ from $[0, 1]$. So even though infinity has lots of weird connotations, there's a precise sense in which (some) results about finite real numbers can be transferred to results about the extended real line, including the infinite points.

Noah Schweber
  • 260,658