Is this a Homeomorphism in Topology?

Question

I want to show a connection between Genotype and Phenotype Spaces, but I need a Homeomorphism from R^1 to R^3 so I can use Binary sequence of funcitons that can map to 3-space. The only problem is I don't remember if R^1 is Homeomorphic to R^3. Sooo.... my question is R^1 homeomorphic to R^3 and can I have a binary map go and map to 3-space?

Answer 1

If there were a homeomorphism $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$, then there would also be a homeomorphism $\mathbb{R}^3 \setminus \{x \} \rightarrow \mathbb{R} \setminus \{\phi(x)\}$ for any $x \in \mathbb{R}^3$ given by the restriction of $\phi$ to that subspace, but this cannot exist because the former is connected whereas the latter is not (continuous maps preserve connectedness). So $\mathbb{R}$ and $\mathbb{R}^3$ cannot be homeomorphic.

As path-connectedness is also preserved under continuous maps, you can also get this result by noting that there exists connected-but-not-path-connected sets in $\mathbb{R}^3$, but not in $\mathbb{R}$ (cf. the topologist's sine curve).

In greater generality, $\mathbb{R}^n$ and $\mathbb{R}^m$ are not homeomorphic whenever $n \neq m$, but showing this requires more sophisticated machinery.