Can the subobject classifier be covered by an object in a subtopos?

Question

Let $\mathcal{E}$ be a topos and $\Omega$ its subobject classifier.

Is it possible to have a nonidentity local operator (a.k.a Lawvere-Tierney topology) $j\colon\Omega\to\Omega$, a $j$-sheaf $X\in\mathcal{E}_j\subseteq\mathcal{E}$, and an epimorphism $f\colon X\twoheadrightarrow\Omega$?

I'd be interested to see an example.

Thanks!

Answer 1

Consider the Sierpinski topos, i.e., presheaves on the poset $P=\{0<1\}$, and consider the topology in which $1$ is covered by $\{0\}$, i.e., the double-negation topology. Then the presheaf $1+1+1$ (i.e., assign to each node of $P$ a 3-element set and let the transition map be a bijection) is a sheaf. It maps surjectively to $\Omega$ by sending its three points to the three global sections of $\Omega$.