  A topologized semilattice $X$ is called complete if each non-empty chain
$C\subset X$ has $\inf C$ and $\sup C$ that belong to the closure $C$ of the
chain $C$ in $X$. In this paper, we introduce various concepts of completeness
of topologized semilattices in the context of operators that generalize the
closure operator, and study their basic properties. In addition, examples of
specific topologized semilattices are given, showing that these classes do not
coincide with each other. Also in this paper, we prove theorems that allow us
to generalize the available results on complete semilattices endowed with
topology.
