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Alon Amit
Alon Amit
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I want to find two closed, non-homeomorphic subsets $A$ and $B$ of $\mathbb{R}$ (with subset topology), with the property that there exist two continuous bijections $$f:A\to B,~~~~g:B\to A.$$

Clearly $A$ or $B$ cannot be bounded. But I didn't find more restrictions. Do we have some results on this question?

I want to find two closed subsets $A$ and $B$ of $\mathbb{R}$ (with subset topology), with the property that there exist two continuous bijections $$f:A\to B,~~~~g:B\to A.$$

Clearly $A$ or $B$ cannot be bounded. But I didn't find more restrictions. Do we have some results on this question?

I want to find two closed, non-homeomorphic subsets $A$ and $B$ of $\mathbb{R}$ (with subset topology), with the property that there exist two continuous bijections $$f:A\to B,~~~~g:B\to A.$$

Clearly $A$ or $B$ cannot be bounded. But I didn't find more restrictions. Do we have some results on this question?

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Karoo Yang
Karoo Yang
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Existence of non-homeomorphic pair of bijectively related closed subsets in $\mathbb{R}$

I want to find two closed subsets $A$ and $B$ of $\mathbb{R}$ (with subset topology), with the property that there exist two continuous bijections $$f:A\to B,~~~~g:B\to A.$$

Clearly $A$ or $B$ cannot be bounded. But I didn't find more restrictions. Do we have some results on this question?