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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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.
J. W. Tanner
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same argument applies for $[0,1]$ and $\Bbb R$, but there are continuous bijections between $(0,1)$ and $\Bbb R$ – J. W. Tanner Jun 26 '19 at 04:23
$\mathbb{R}$gives $\mathbb{R}$. – Nosrati Jun 26 '19 at 03:45