I apologize if my question is trivial. I am a group theorist with a minor knowledge of topology. Suppose $(X, T\_X)$ and $(Y, T\_Y)$ are two topological spaces and there is an inclusion-reversing (inclusion preserving) bijection $\varphi:T\_X\to T\_Y$. What is the name of this kind of equivalence? What can be said about the relation of these two spaces?



What is the name of this kind of equivalence between two topological spaces?
As far as I can see there are two possibilities where we can say more:


* The bijection is order preserving. As Qiaochu Yuan said in a comment, this is the same as asking that their associated [locales](https://ncatlab.org/nlab/show/locale) are isomorphic. Thanks to the [equivalence](https://en.wikipedia.org/wiki/Stone_duality) of sober spaces with spatial locales, this is the same as asking that their soberifications are homeomorphic. In particular, if $X$ and $Y$ are already sober (a very common condition, sober is strictly weaker than Hausdorff), this is the same as asking that $X$ and $Y$ are homeomorphic.
* The bijection is order reversing. Then the lattice of opens of both spaces need to have arbitrary meets distribute over finite joins (i.e. each [frame](https://en.wikipedia.org/wiki/Complete_Heyting_algebra) of opens must have the property that its opposite lattice is a frame too). This is a very rare situation, that happens mainly for [Alexandrov spaces](https://en.wikipedia.org/wiki/Alexandrov_topology),although as მამუკაჯიბლაძე notes in the comments there might be other examples.


