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?



Existence of non-homeomorphic pair of bijectively related closed subsets in $\mathbb{R}$
Using a relatively recent theorem, we can get a whole family of examples.


Suppose $A$ and $B$ are two closed subsets of the real line satisfying the following properties:


* zero-dimensional
* $\sigma$-compact, but not compact
* no isolated points


Then they are "bijectively related" in the sense of your question. A proof can be found in Section 3 of this paper: <https://wrbrian.files.wordpress.com/2012/01/cumet2.pdf>.


It can be shown that there are $2^{\aleph\_0}$ homeomorphism types of spaces satisfying these conditions (though, sadly, I don't have a reference). So now you have uncountably many examples!


