  Let $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ be the bicyclic semigroup
extension for the family $\mathscr{F}$ of ${\omega}$-closed subsets of $\omega$
which is introduced in \cite{Gutik-Mykhalenych=2020}. We study topologizations
of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ for the family
$\mathscr{F}$ of inductive ${\omega}$-closed subsets of $\omega$. We generalize
Eberhart-Selden and Bertman-West results about topologizations of the bicyclic
semigroup \cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every
Hausdorff shift-continuous topology on the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is discrete and if a Hausdorff
semitopological semigroup $S$ contains $\boldsymbol{B}_{\omega}^{\mathscr{F}}$
as a proper dense subsemigroup then
$S\setminus\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is an ideal of $S$. Also, we
prove the following dichotomy: every Hausdorff locally compact shift-continuous
topology on $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with an adjoined zero is
either compact or discrete. As a consequence of the last result we obtain that
every Hausdorff locally compact semigroup topology on
$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with an adjoined zero is discrete and
every Hausdorff locally compact shift-continuous topology on the semigroup
$\boldsymbol{B}_{\omega}^{\mathscr{F}}\sqcup I$ with an adjoined compact ideal
$I$ is either compact or the ideal $I$ is open, which extent many results about
locally compact topologizations of some classes of semigroups onto extensions
of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$.
