# Does there exist any continuous bijection between [0,1] and (0,1) and between [0,1] and IR? [duplicate]

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.

• I do not know how to use mathjax,so please someone edit my question. – user684834 Jun 26 '19 at 3:43
• What is IR?${}{}$ – Clayton Jun 26 '19 at 3:43
• Put them between dollars. $\mathbb{R}$ gives $\mathbb{R}$. – Nosrati Jun 26 '19 at 3:45
• Note: $[0,1]$ is compact and $(0,1)$ is not – J. W. Tanner Jun 26 '19 at 3:46
• @Clayton I think it's interval $\mathbb R$. – Michael Rozenberg Jun 26 '19 at 3:48

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.
• assuming the standard topology for $\Bbb R$ – J. W. Tanner Jun 26 '19 at 4:06
• 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 4:23