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Consider the following equivalence relation on topological spaces:

$X\sim Y$ $:\Longleftrightarrow$ there are bijective continuous maps $\phi:X\to Y$ and $\psi:Y\to X$.

Note that there are no requirements on $\phi\circ\psi$ and $\psi\circ\phi$. Is there a name for this relation, the equivalence classes or the study of topological spaces up to this relation? Any keywords? Is there anything general that can be said about spaces that fall in the same class? For example, if $X\sim Y$ then these spaces cannot be distinguished by what embeds into them and where they embed into. Note that $X\sim Y$ does not imply that the spaces are homeomorphic.

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    $\begingroup$ What is the easiest example of two such nonhomeomorphic spaces ? $\endgroup$
    – HenrikRüping
    Commented Feb 16 at 12:58
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    $\begingroup$ Is there an example with finite CW complexes? $\endgroup$
    – Sam Hopkins
    Commented Feb 16 at 14:31
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    $\begingroup$ More examples on MSE: math.stackexchange.com/questions/4461553/… and math.stackexchange.com/questions/4486360/… $\endgroup$
    – Mark Grant
    Commented Feb 16 at 16:22
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    $\begingroup$ @SamHopkins: finite CW complexes are compact hausdorff. It is a theorem that if $X$ is compact and $Y$ Hausdorff, then any continuous bijection $\phi:X\to Y$ is a homeomorphism. $\endgroup$
    – Willie Wong
    Commented Feb 16 at 23:07
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    $\begingroup$ @HenrikRüping: How about $X = $ the Baire space and $Y = $ a disjoint union of the Baire space and the Cantor space. The Baire space is nowhere compact, so $X$ and $Y$ are not homeomorphic. But there is a continuous bijection $X \rightarrow Y$ (Baire space is homeomorphic to two copies of itself, and you can find a continuous bijection from one of these copies onto Cantor space) and from $Y \rightarrow X$ (roughly, if you excise a copy of Cantor space from Baire space, what's left is Baire space; doing this in reverse gives you the required map). $\endgroup$
    – Will Brian
    Commented Feb 17 at 18:51

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This relation was introduced (I don't know if for the first time) in the 1984 paper Bijectively related spaces I: Manifolds by P. H. Doyle and J. G. Hocking. As the title indicates, two spaces that are equivalent with respect to your relation are called bijectively related.

You can find more information in this question of mine which was motivated by this question of Henno Brandsma.

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  • $\begingroup$ As an aside: you indicated in your previous question that $br(X) = 1$ for $X$ compact. Were you assuming Hausdorff? If not, how did you argue for the non-Hausdorff case? $\endgroup$
    – Willie Wong
    Commented Feb 17 at 2:59
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    $\begingroup$ An even earlier reference: Sierpinski discusses this equivalence relation in his 1956 book General Topology (second edition). He defines two spaces to be of the same $\gamma$-type if they are related in this way. I don't have the book with me right now, and I don't recall what, if anything, he proves about spaces having the same $\gamma$-type. $\endgroup$
    – Will Brian
    Commented Feb 17 at 18:46
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    $\begingroup$ To follow up on Will Brian's comment, the relevant section of the book is Chapter VI, Section 62, "Biuniform and continuous images of sets." He devotes about a page and a half to this subject, mostly quoting results from the literature. One of the main references is Sur les images biunivoques et continues dans un sens, by Wacław Sierpiński, Fundamenta Mathematicae 26 (1936), 44-49. $\endgroup$
    – Timothy Chow
    Commented Feb 21 at 19:48
  • $\begingroup$ @WillieWong, I was probably assuming Hausdorff (I´ll edit the post to make it clear) since you can easily modify any pair of bijectively related non-homeomorphic spaces to make them compact albeit non-Hausdorff while keeping them biyectively related and non-homeomorphic. $\endgroup$
    – Ramiro de la Vega
    Commented Feb 24 at 20:20

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