# Show there is no continuous injective maps R^2 to R

• Oct 1st 2008, 04:33 AM
szpengchao
Show there is no continuous injective maps R^2 to R
Show there is no continuous injective maps : R^2----> R.
• Oct 20th 2008, 02:44 PM
Rebesques
Suppose there is.

We know that $\mathbb{R}^2$ is homeomorphic to the two dimensional sphere minus a point, $\mathbb{S}^2 - \{a\}$, and that $\mathbb{R}$ is homeomorphic to the circle minus a point, $\mathbb{S}^1 - \{x\}$.
So we have a continuous and injective map $f:\mathbb{S}^2 - \{a\} \rightarrow \mathbb{S}^1 - \{x\}$.

Remove a point $f(b)=y\neq x$ from $f(\mathbb{S}^2 - \{a\})$.

Choose two points $p=f(c), q=f(d) \in \mathbb{S}^1-\{x\}$, such that $x$ is contained in the arc to join $p$ and $q$. There exists a simple curve $\gamma$ on $\mathbb{S}^2-\{a,b\}$ to connect $c, d$. Then, $f(\gamma)$ must be a continuous simple curve to connect $p,q$ on $f(\mathbb{S}^2-\{a,b\})$. This is a contradiction as $p,q$ belong to two disjoint connected components of $f(\mathbb{S}^2-\{a,b\})$.