  In this article the notions of (quasi weakly hereditary) general closure
operator $\mb{C}$ on a category $\cx$ with respect to a class $\cm$ of
morphisms, and quasi factorization structures in a category $\cx$ are
introduced. It is shown that under certain conditions, if $(\ce, \cm)$ is a
quasi factorization structure in $\cx$, then $\cx$ has quasi right
$\cm$-factorization structure and quasi left $\ce$-factorization structure. It
is also shown that for a quasi weakly hereditary and quasi idempotent
QCD-closure operator with respect to a certain class $\cm$, every quasi
factorization structure $(\ce, \cm)$ yields a quasi factorization structure
relative to the given closure operator; and that for a closure operator with
respect to a certain class $\cm$, if the pair of classes of quasi dense and
quasi closed morphisms forms a quasi factorization structure, then the closure
operator is both quasi weakly hereditary and quasi idempotent. Several
illustrative examples are furnished.
