  One of the main prerequisites for understanding sheaves on elementary toposes
is the proof that a (Lawvere-Tierney) topology on a topos induces a closure
operator on it, and vice-versa. That standard theorem is usually presented in a
relatively brief way, with most details being left to the reader and with no
hints on how to visualize some of the hardest axioms and proofs.
  These notes are, on a first level, an attempt to present that standard
theorem in all details and in a visual way, following the conventions in "On my
favorite conventions for drawing the missing diagrams in Category Theory"
[Ochs2020]; in particular, some properties, like stability by pullbacks, are
always drawn in the same "shape".
  On a second level these notes are also an experiment on doing these proofs on
"archetypal cases" in ways that makes all the proofs easy to lift to the
"general case". Our first archetypal case is a "topos with inclusions". This is
a variant of the "toposes with canonical subobjects" from [Lambek/Scott 1986];
all toposes of the form $\mathbf{Set}^\mathbf{C}$, where $\mathbf{C}$ is a
small category, are toposes with inclusions, and when we work with toposes with
inclusions concepts like subsets and intersections are very easy to formalize.
We do all our proofs on the correspondence between closure operators and
topologies in toposes with inclusions, and then we show how to lift all our
proofs to proofs that work on any topos. Our second archetypal case is toposes
of the form $\mathbf{Set}^\mathbf{D}$, where $\mathbf{D}$ is a finite
two-column graph. We show a way to visualize all the LT-topologies on toposes
of the form $\mathbf{Set}^\mathbf{D}$, and we define formally a way to "add
visual intuition to a proof about toposes"; this is related to the several
techniques for doing "Category Theory for children" that are explained in "On
my favorite conventions...".
