Interesting Lawvere-Tierney topologies on Set

Question

When doing topos theory I like working out hard theorems for Set first, and then translating back to general topoi. For stuff like subobjects and regular epi-mono factorization this works great, but I have not been able to think of or find on google a Lawvere-Tierney topology on Set except for the identity on $\Omega$. Are there any interesting Lawvere-Tierney topologies on Set?

Answer 1

If we look at the axioms of a Lawvere-Tierney topology (see e.g. nLab), then in $\mathbf{Set}$ this is a map $j: \{0,1\} \to \{0,1\}$ such that (for any $x,y \in \{0,1\}$):

1. $j(1) = 1$,
2. $jj(x) = j(x)$,
3. $j(x \wedge y) = j(x) \wedge j(y)$.

There are clearly only two options for $j$: the identity on $\{0,1\}$ as you mentioned and the map $j(0) = j(1) = 1$. So no, there are no other interesting Lawvere-Tierney topologies.