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 11 function f:XY and another 11 function g:YX. Then there exists a 11, onto function h:XY and hence XY .





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



Define g:(0,1)×(0,1)(0,1) by g(x,y)=0.x1y1x2y2....
where x=0.x1x2x3.... and y=0.y1y2y3.... and where we make the convention that we always use the terminating form over the repeating 9s form when the situation arises.



Then g is injective. (Prove this!)



Hence (0,1)(0,1)×(0,1)




We know (0,1) and R are equivalent,via xtanπ(2x1)/2



So, next map R2 bijectively onto the open unit square (0,1)×(0,1) by mapping each R bijectively onto the open interval (0,1)



Hence (0,1)×(0,1)R2




Summary:



(0,1)(0,1)×(0,1)R2





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


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