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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."

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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 about it, and some slides at his web-page.

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