Can a finite amount of surface be in any way folded to a infinite amount of surface area or length?

Question

Exactly what the title says. There's plenty of mathematical objects that have infinite surface area or length because we start out with the assumption of their infinite characteristic. But, is it possible through any kind of continuous topological transformation to bend, say, a finite line segment so that it has infinite length?

Answer 1

Yes. Basically, continuity can't tell the difference between a shape being "finite but open on the ends" and "infinite". For an explicit example, consider the function $f:(-1,1)\to\mathbb{R}$ given by $f(x)=\frac{x}{x^2-1}$. This is a continuous bijection (in fact, a homeomorphism) between the finite interval $(-1,1)$ and the infinite line $\mathbb{R}$. As far as topology is concerned, as you approach the endpoints $-1$ and $1$ of the interval $(-1,1)$, you might as well instead be going out to infinity.

Note that it is crucial here that we have an open interval $(-1,1)$. If you have a closed interval such as $I=[-1,1]$ instead, then $I$ is compact and so its image under any continuous map is also compact. In particular, if $f:I\to \mathbb{R}$ is any continuous map, then the image of $f$ is bounded. (However, if we take the extended real line $[-\infty,\infty]=\mathbb{R}\cup\{-\infty,\infty\}$, then we still can get a homeomorphism $[-1,1]\to[-\infty,\infty]$ by extending the $f$ of the previous paragraph to the endpoints.)