0
$\begingroup$

I need a reference on theorems unique to $\mathbb{R}$, things that go away in higher dimensions.

For example: in Topology and Groupoids, it is said that a continuous injective function $f: (a, b) \to \mathbb{R}$ is a homeomorphism to its image. I completely forgot about this fact, yet it is hugely important for gaining better intuitition for what topology of $\mathbb{R}$ is like. Does it still hold in higher dimensions? No, the 8-curve is a famous counter-example.

  • 0
    Doesn't it? I would say that a higher-dimensional analogue would be: any continuous injection of a ball in ${\bf R}^n$ into ${\bf R}^n$ is an embedding. This looks true for me (can't think of a formal proof right now, though...).2012-11-04
  • 0
    @tomasz Yeah, we can check continuity of the inverse by probing it with curves.2012-11-04

1 Answers 1