  This paper deals with the notion of weak Lawvere-Tierney topology on a topos.
Our motivation to study such a notion is based on the observation that the
composition of two Lawvere-Tierney topologies is no longer idempotent, when
seen as a closure operator. For a given topos $\mathcal{E}$, in this paper we
investigate some properties of this notion. Among other things, it is shown
that the set of all weak Lawvere-Tierney topologies on $\mathcal{E}$
constitutes a complete residuated lattice provided that $\mathcal{E}$ is
(co)complete. Furthermore, when the weak Lawvere-Tierney topology on
$\mathcal{E}$ preserves binary meets we give an explicit description of the
(restricted) associated sheaf functor on $\mathcal{E}$.
