Shape? At the usual mathematical literature when we can discuss about the shape of a "space" that we have a kind of "topography" on it. For example a topology, metric, geometry, etc.
Note that for example: we can define an "unformal" topology on proper class of $Ord$ which any continuous map $f:Ord\longrightarrow Ord$ in the sense of $\forall \alpha\in limord~~~f(\alpha)=\bigcup\_{\beta \in \alpha} f(\beta)$ be continuous in the sense of this topology. Now the main question is:


**Main Question:** Is there any known "non trivial" metric, topology,... on the proper class of all sets or other proper classes like $HOD$, $L$?


**Remark:** Non trivial in the above question means "a topology, metric,... which the proof of being a topology or metric needs an essential use of the special properties of the ground proper class and this definition doesn't work for every collection of sets."



What is the shape of mathematical universe?
There is an answer to this for the universe of constructive mathematics. According to Martín Escardó, a universe of Martin-Löf type theory has indiscrete topology, constructively. He has a [draft](http://www.cs.bham.ac.uk/~mhe/papers/universe-indiscrete-and-rice.pdf) about it, and [some slides](http://www.cs.bham.ac.uk/~mhe/.talks/4WFT/escardo-4WFT.pdf) at his web-page.


