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We know that there are bijections between $[0,1]$, $(0,1)$ and $\mathbb{R}$. But my question is can we obtain a continuous bijection between $[0,1]$ and $(0,1)$, and between $[0,1]$ and $\mathbb{R}$? I think there will not exist but I am not sure.

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  • $\begingroup$ I do not know how to use mathjax,so please someone edit my question. $\endgroup$ – user684834 Jun 26 '19 at 3:43
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    $\begingroup$ What is IR?${}{}$ $\endgroup$ – Clayton Jun 26 '19 at 3:43
  • $\begingroup$ Put them between dollars. $\mathbb{R}$ gives $\mathbb{R}$. $\endgroup$ – Nosrati Jun 26 '19 at 3:45
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    $\begingroup$ Note: $[0,1]$ is compact and $(0,1)$ is not $\endgroup$ – J. W. Tanner Jun 26 '19 at 3:46
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    $\begingroup$ @Clayton I think it's interval $\mathbb R$. $\endgroup$ – Michael Rozenberg Jun 26 '19 at 3:48
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The image of a continuous map of a compact metric space is compact. In particular, the image of a continuous map of a compact metric space into $\mathbb R$ is closed and bounded. Therefore, don't expect to find a continuous bijection between $[0,1], $ which is compact, and $(0,1)$, which is open. Other explanations can be found here.

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  • $\begingroup$ assuming the standard topology for $\Bbb R$ $\endgroup$ – J. W. Tanner Jun 26 '19 at 4:06
  • $\begingroup$ same argument applies for $[0,1]$ and $\Bbb R$, but there are continuous bijections between $(0,1)$ and $\Bbb R$ $\endgroup$ – J. W. Tanner Jun 26 '19 at 4:23