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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 aBanach space $E$ is DNS, if no restriction of $T$ to a subspace generated by adisjoint 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 thatin this case $T$ is DNS if and only if the norm topology is the minimaltopology which is simultaneously stronger than the unbounded norm topology andthe topology generated by $T$ as a map (we say that $T$ "complements" theunbounded norm topology in $F$). Since the class of DNS operators plays asimilar role in the category of Banach lattices as the upper semi-Fredholmoperators play in the category of Banach spaces, we investigate and indeeduncover a similar characterization of the latter class of operators, but thistime they have to complement the weak topology.

Disjointly non-singular operators on order continuous Banach lattices complement the unbounded norm topology