For *weakly cohesive toposes*, there exists a notion of *contractability*, and toposes with a subobject classifier $\Omega$ that is contractible are of special interest (see [here](https://ncatlab.org/nlab/show/sufficiently+cohesive+topos#cohesively_connected_truth)).


It occured to me that one could view probability theory as a theory of (convex) interpolation of truth, which replaces boolean truth $\Omega = \{0,1\}$ by 'continuous truth' $\Omega = [0,1]$, so I wonder, is there an example of a topos with a subobject classifier given (in a suitable sense) by the real unit interval?



Topos with $\Omega = [0,1]$?
Here's an example which, I'm afraid, is not very interesting (and may not match your notion of “suitable sense”): let $X$ be the topological space $\mathopen]0,1\mathclose[$ (the open interval) with the (not at all separated) topology given by the $U\_x := \mathopen]0,x\mathclose[$ for $x \in [0,1]$ (with $U\_0 = \varnothing$, obviously). Since $U\_x \cap U\_y = U\_{\min(x,y)}$ and
$$
\bigcup\_{i\in I} U\_{x\_i} = U\_{\sup\{x\_i : i\in I\}}
$$
this is indeed a topology. Of course, $\mathcal{O}(X) := \{U\_x : x\in[0,1]\}$ can be identified with $[0,1]$ as an ordered set (indeed, frame). The topos of sheaves on $(X, \mathcal{O}(X))$ then has this as subobject classifier.


