On topologization of the extended bicyclic semigroup

Non-discrete semigroup $T_1$-topologies on the extended bicyclic semigroup ${\mathcal{C}}_{\mathbb{Z}}$ are constructed. Also, we present topological conditions, when a semigroup (shift-continuous) $T_1$-topology on ${\mathcal{C}}_{\mathbb{Z}}$ is discrete.