Saturday, 19 January 2019

real analysis - Is there a bijection between mathbbR2 and (0,1)?




True or false: there exists a bijection between R2 and the open interval (0,1).




I think this is false, because R2{0} is connected but (0,1)=R as R{0} is not connected, as the continuous image of a connected set is connected.


Answer




The answer is True.



At this time I cannot provide the explicit bijection BUT the idea is this:




Schroder–Bernstein Theorem:



Assume there exists a 1–1 function f : X \rightarrow Y and another 1–1 function g : Y \rightarrow X. Then there exists a 1–1, onto function h : X \rightarrow Y and hence X ∼ Y .





Define f:(0,1) \rightarrow (0,1)\times (0,1) by f(x)=(x,1/3)
Then f is injective.



Define g:(0,1)\times (0,1) \rightarrow (0,1) by g(x,y)=0.x_1y_1x_2y_2....
where x=0.x_1x_2x_3.... and y=0.y_1y_2y_3.... and where we make the convention that we always use the terminating form over the repeating 9's form when the situation arises.



Then g is injective. (Prove this!)



Hence (0,1) \sim (0,1) \times (0,1)




We know (0,1) and \Bbb{R} are equivalent,via x \mapsto \tan \pi (2x-1)/2



So, next map \Bbb{R}^2 bijectively onto the open unit square (0, 1)\times (0,1) by mapping each \Bbb{R} bijectively onto the open interval (0, 1)



Hence (0, 1)\times (0,1) \sim \Bbb{R}^2




Summary:



(0,1) \sim (0,1) \times (0,1) \sim \Bbb{R}^2





In addition, if your function is continuous, then this is not true!


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