Are R^2 and R^n Homeomorphic If n is Not Equal to 2?

[gop]

Homework Statement

Prove that $$R^2$$ and $$R^n$$ are not homeomorphic if $$n\neq2$$ (Hint: Consider the complement of a point in $$R^2$$ or $$R^n$$).

Homework Equations

The Attempt at a Solution

The proof that $$R^n$$ is not homeomorphic to $$R$$ is done by considering that if they are homeomorphic i.e. there exists a continuous bijection with continuous inverse $$f:R\rightarrow R^n$$. Then the restriction $$f:R\backslash {0} \rightarrow R^n\backslash f(0)$$ is also a continuous bijection. Since $$R\backslash {0}$$ is not connected but $$R^n\backslash f(0)$$ is if $$n\neq 1$$ we have a contradiction.

However, to show that $$R^2$$ is not homeomorphic to $$R^n$$ this doesn't work. There is also no other topological invariant I could detect. Both $$R^2$$ and $$R^n$$ are contractible to a point and thus have the same fundamental group, for example.

I would appreciate any idea
thx

 

[CompuChip]

Instead of taking out a point, try taking out something bigger :)

 

[HallsofIvy]

Homework Statement

Prove that $$R^2$$ and $$R^n$$ are not homeomorphic if $$n\neq2$$ (Hint: Consider the complement of a point in $$R^2$$ or $$R^n$$).

Homework Equations

The Attempt at a Solution

The proof that $$R^n$$ is not homeomorphic to $$R$$ is done by considering that if they are homeomorphic i.e. there exists a continuous bijection with continuous inverse $$f:R\rightarrow R^n$$. Then the restriction $$f:R\backslash {0} \rightarrow R^n\backslash f(0)$$ is also a continuous bijection. Since $$R\backslash {0}$$ is not connected but $$R^n\backslash f(0)$$ is if $$n\neq 1$$ we have a contradiction.

However, to show that $$R^2$$ is not homeomorphic to $$R^n$$ this doesn't work. There is also no other topological invariant I could detect. Both $$R^2$$ and $$R^n$$ are contractible to a point and thus have the same fundamental group, for example.

I would appreciate any idea
thx

You can continue that idea. In R²\{0}, a closed loop containing a point p cannot be contracted to a point. In Rⁿ, for n larger than 2, it can.