Show there is no continuous injective maps : R^2----> R.

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- Oct 1st 2008, 03:33 AMszpengchaoShow there is no continuous injective maps R^2 to R
Show there is no continuous injective maps : R^2----> R.

- Oct 20th 2008, 01:44 PMRebesques
Suppose there is.

We know that $\displaystyle \mathbb{R}^2$ is homeomorphic to the two dimensional sphere minus a point, $\displaystyle \mathbb{S}^2 - \{a\}$, and that $\displaystyle \mathbb{R}$ is homeomorphic to the circle minus a point, $\displaystyle \mathbb{S}^1 - \{x\}$.

So we have a continuous and injective map $\displaystyle f:\mathbb{S}^2 - \{a\} \rightarrow \mathbb{S}^1 - \{x\}$.

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

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