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→Y and another 1–1 function g:Y→X. Then there exists a 1–1, onto function h:X→Y and hence X∼Y .
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 9′s 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 x↦tanπ(2x−1)/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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