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I want a graphical interpretation of a bijection between a close bounded interval and the real line.Can someone give me a bijection not explicitly but graphically.

  • It's a lot easier to visualize a bijection between $[0,1]$ and $(0,1)$, rather than the entire number line. Would that be enough? – Arthur Jun 26 '19 at 12:44
  • Tell me how to visualize that between [0,1] and (0,1).But it would not be enough for me because I want to visualize a bijection between [0,1] and R. –  Jun 26 '19 at 15:07

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There are two steps. The first is showing a bijection from an open interval to $\mathbb{R}$ and the second is to show a bijection from an open interval onto a closed interval.

A bijection from $(-\pi/2, \pi/2)$ to $\mathbb{R}$ is given by $\tan(x)$.

For the second part: How to define a bijection between $(0,1)$ and $(0,1]$?

EDIT: There are other options for the first bijection here: Is there a bijective map from $(0,1)$ to $\mathbb{R}$?

twnly
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  • You did not get my question I think,I want a bijection that can be visualized I mean some diagram showing a one-one correspondence. –  Jun 27 '19 at 15:28