  In this article we investigate the disjointly non-singular (DNS) operators.
Following [8] we say that an operator $T$ from a Banach lattice $F$ into a
Banach space $E$ is DNS, if no restriction of $T$ to a subspace generated by a
disjoint sequence is strictly singular. We partially answer a question from [8]
by showing that this class of operators forms an open subset of
$L\left(F,E\right)$ as soon as $F$ is order continuous. Moreover, we show that
in this case $T$ is DNS if and only if the norm topology is the minimal
topology which is simultaneously stronger than the unbounded norm topology and
the topology generated by $T$ as a map (we say that $T$ "complements" the
unbounded norm topology in $F$). Since the class of DNS operators plays a
similar role in the category of Banach lattices as the upper semi-Fredholm
operators play in the category of Banach spaces, we investigate and indeed
uncover a similar characterization of the latter class of operators, but this
time they have to complement the weak topology.
