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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.

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    @tomasz Yeah, we can check continuity of the inverse by probing it with curves.2012-11-04

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This answer is wrong as shown by the OP, but I thought his comment and the link he gave are valuable, so I leave this answer intact:

This isn't really a big theorem, but in $\mathbb R$, if the left hand and the right hand limits exist and are equal, then the limit exists. This idea dissipates once you go to higher dimensions (even just $\mathbb R^2$). There's no more left and right limits, you need to approach the limit point through any path. With that being said, maybe there's a theorem that states that if all those limits exist, then the limit exists. I don't know how useful that theorem would be though.

I tried thinking of more substantial examples before submitting, but couldn't find any. I haven't necessarily seen every single-variable theorem generalized to higher dimensions, but the few I thought of seem to be easily generalized. I'm intrigued now to see what are some other, bigger results that can't be generalized.

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    Thank you! I stand corrected. However, I do stand behind my assertion that the $\Longleftarrow$ part of the theorem isn't the most useful. That is, there's no way of showing that the limit along all paths is the same (or is there and I'm not aware of it?), so it doesn't help us in proving the existence of a limit. It does help us disprove the existence of the limit, but that can be done with the $\Longrightarrow$ part of the theorem. I'm not saying that this theorem should be thrown out or shortened, just merely thinking out loud about the usefulness of one half of it.2012-11-04