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 is homeomorphic to the two dimensional sphere minus a point, , and that is homeomorphic to the circle minus a point, .

So we have a continuous and injective map .

Remove a point from .

Choose two points , such that is contained in the arc to join and . There exists a simple curve on to connect . Then, must be a continuous simple curve to connect on . This is a contradiction as belong to two disjoint connected components of .