On a semitopological semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ when a family $\mathscr{F}$ consists of inductive non-empty subsets of $\omega$

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