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Math Help - Show there is no continuous injective maps R^2 to R

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

    Show there is no continuous injective maps : R^2----> R.
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  2. #2
    Super Member Rebesques's Avatar
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    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\}).
    Last edited by Rebesques; October 20th 2008 at 01:55 PM.
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