  In the paper we introduce topological $\mathbb{Z}$-Bruck-Reilly and
topological $\mathbb{Z}$-Bruck extensions of (semi)topological monoids which
are generalizations of topological Bruck-Reilly and topological Bruck
extensions of (semi)topological monoids and study their topologizations. The
sufficient conditions under which the topological $\mathbb{Z}$-Bruck-Reilly
($\mathbb{Z}$-Bruck) extension admits only the direct sum topology and
conditions under which the direct sum topology can be coarsened are given.
Also, topological characterizations of some classes of $I$-bisimple
(semi)topological semigroups are given.
