  Let $j$ be a Lawvere-Tierney topology (a topology, for short) on an arbitrary
topos $\mathcal{E}$, $B$ an object of $\mathcal{E}$, and $j_B = j\times 1_B$
the induced topology on the slice topos $\mathcal{E}/B$. In this manuscript, we
analyze some properties of the pullback functor $\Pi_B:\mathcal{E}\rightarrow
\mathcal{E}/B$ which have deal with topology. Then for a left cancelable class
$\mathcal{M}$ of all $j$-dense monomorphisms in a topos $\mathcal{E}$, we
achieve some necessary and sufficient conditions for that $(\mathcal{M} ,
\mathcal{M}^{\perp})$ is a factorization system in $\mathcal{E}$, which is
related to the factorization systems in slice topoi $\mathcal{E}/B,$ where $B$
ranges over the class of objects of $\mathcal{E}$.
  Among other things, we prove that an arrow $f : X\rightarrow B$ in
$\mathcal{E}$ is a $j_B$-sheaf whenever the graph of $f$, is a section in
$\mathcal{E}/B$ as well as the object of sections $S(f)$ of $f$, is a $j$-sheaf
in $\mathcal{E}$. Furthermore, we introduce a class of monomorphisms in
$\mathcal{E}$, which we call them $j$-essential.
  Some equivalent forms of those and some of their properties are presented.
Also, we prove that any presheaf in a presheaf topos has a maximal essential
extension. Finally, some similarities and differences of the obtained result
are discussed if we put a (productive) weak topology $j$, studied by some
authors, instead of a topology.
