TITLE: What is a Lawvere-Tierney topology?
QUESTION [5 upvotes]: I've read some articles and books for the definition and use of Lawvere-Tierney topologies, but I still don't understand their role.
Some people introduce these topologies as modal operators for logic. Others speak about them as a way to define "locally true" propositions, although I don't see what it means.
When others introduce them as topologies on a category, I don't understand if it allows to define open sets on the category or on the image sets by any presheaf.
Could someone give me some clarifications about this ? In particular I'd like to see some examples of them in simple cases (I've been working with the Sierpinsky topos for example) and their use and advantage.

REPLY [3 votes]: I'm going to give an example, which works on any topos, and therefore also on toposes of sheaves on topological spaces.
Double negation $\neg\neg$ is a local operator or Lawvere-Tierney topology. It acts as follows on the lattice of open subsets of a topological space $X$: for each open set $U$, $\neg\neg U$ is the interior of the closure of $U$. To every open set $U$ the operator joins all isolated subsets of the closed complement of $U$. Open sets that are stable for this topology (this means $\neg\neg U=U$) are known as regular open subsets of the space.
Lawvere-Tierney topologies in general redefine what an open cover is, without changing much about open sets themselves. In our example a family of open sets $\{U_i|i\in\kappa\}$ covers $X$ if $X=\neg\neg (\bigcup_{i\in\kappa} U)$, i.e., we ignore it when covers miss isolated closed subsets of $X$.
The lattice of regular open subsets is always a Boolean algebra, and one of the applications of $\neg\neg$ is in constructing models for classical theories. In fact, $\neg\neg$ is part of the construction of Cohen's counterexample to the continuum hypothesis (see MacLane and Moerdijk's "Sheaves in  Geometry and Logic" for this).