On the extended bicyclic semigroup $\mathscr{C}_\mathbb{Z}$ the following non-discrete $T_1$-topologies are constructed: $(i)$ a semigroup inverse topology; $(ii)$ a Baire left-continuous [right-continuous] topology; $(iii)$ a locally compact left-continuous [right-continuous] topology. We prove that, on $\mathscr{C}_\mathbb{Z}$ does not exist a countably compact left-continuous [right-continuous] $T_1$-topology. We give conditions under which a $T_1$-topology on $\mathscr{C}_\mathbb{Z}$ is discrete. In particular, $(i)$ a left-continuous [right-continuous] $T_1$-topology $\tau$ on $\mathscr{C}_\mathbb{Z}$ is discrete if and only if there exists a sequence $\{(i_n,j_n)\}_{n\in\mathbb{N}}$ of isolated points in $(\mathscr{C}_\mathbb{Z},\tau)$ such that the sequence $\{i_n\}_{n\in\mathbb{N}}$ $[\{j_n\}_{n\in\mathbb{N}}]$ is strictly increasing; $(ii)$ a shift-continuous $T_1$-topology $\tau$ on $\mathscr{C}_\mathbb{Z}$ is discrete if and only if the space $(\mathscr{C}_\mathbb{Z},\tau)$ contains an isolated point; $(iii)$ if the $T_1$-space of a topological inverse semigroup $(\mathscr{C}_\mathbb{Z},\tau)$ has a point $x$ such that $(\mathscr{C}_\mathbb{Z},\tau)$ is quasi-regular at $x$, then $\tau$ is discrete.

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