Isomorphic between plane and line

[hermitian]

Hi,

I understand that the open interval (0,1) is isomorphic to the real line \mathbb{R}^1

May i know whether there is also isomorphism from \mathbb{R}^2 to \mathbb{R}^1

thanks a lot!

 

[Dragonfall]

What do you mean by an isomorphism in this case?

 

[hermitian]

hmm... the usual sense of isomorphism... where there is a structure preserving bijective map from R^2 to R^1. sorry is my question still vague?

 

[Hurkyl]

The question is which structure are you interested in! It's structure as a set? As an additive group? As a topological space?

I imagine it's the topology you care about. In that case, the answer is "no". The common proof (AFAIK) is to consider the image of a subset of R² in the shape of the letter Y -- if such an isomorphism existed, it must be injective on the Y. Can you derive a contradiction from that?

 

[g_edgar]

There is a Borel isomorphism between \mathbb{R}^2 and \mathbb{R}.

There is a group somorphism between \mathbb{R}^2 and \mathbb{R} (both with addition).

BUT: There is no map between \mathbb{R}^2 and \mathbb{R} that is both a Borel isomorphism and a group isomorphism.