# Math Help - Homeomorphism

1. ## Homeomorphism

How can I prove that every subset of R^n is homeomorphic to a closed subset of R^(n+1)? Does this only works for real number only as R is metrizable?

Thanks.

2. Hello.

I haven't thought about the problem too deeply or anything, but it seems that the map $f:\mathbb{R}^n\to \mathbb{R}^{n+1}$ defined by $f(x_1,\ldots,x_n)=(x_1,\ldots,x_n,0)$ would do the trick.

Did you try it?

3. But the inverse of f may not be continuous unless we restrict the range of inverse of f to the set in R^(n+1).

4. Isn't that what we're doing? We want a homeomorphism of $\mathbb{R}^n$ into some subset of $\mathbb{R}^{n+1}$. In this case, we implicitly chose the subset $\{(x_1,\ldots,x_n,0)\in \mathbb{R}^{n+1}\ |\ x_1,\ldots,x_n\in \mathbb{R}\}$.