.to construct a bijection

between [0,1] and [0,1)......................

I just think it is impossible , cuz the second one is one element less than the first....

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- September 11th 2009, 03:08 AMfelixmcgradyBijection
.to construct a bijection

between [0,1] and [0,1)......................

I just think it is impossible , cuz the second one is one element less than the first.... - September 11th 2009, 04:56 AMputnam120
Well by that logic it would then be impossible to construct a bijection between and .

- September 11th 2009, 05:28 AMTaluivren
Hi,

It must be possible.

If you know Cantor-Bernstein Theorem, it is easy, because there obviously exist an injective map from [0,1) to [0,1] (the identity map) and an injective map from [0,1] to [0,1) (x -> x/2).

If you don't know the theorem, consider set and set . You can construct bijection by setting (verify it is a bijection).

Next, we see that , so the identity on is a bijection from to .

Now we're ready to construct a bijection as follows: