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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