On a semitopological semigroup B_ω^F when a family F consists of inductive non-empty subsets of ω, volume=59, url=http://www.matstud.org.ua/ojs/index.php/matstud/article/view/373, DOI=10.30970/ms.59.1.20-28, abstractNote= Let B_ω^F be the bicyclic semigroup extension for the family F of ω-closed subsets of ω which is introduced in Gutik-Mykhalenych=2020. We study topologizations of the semigroup B_ω^F for the family F of inductive ω-closed subsets of ω. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup Bertman-West-1976, Eberhart-Selden=1969 and show that every Hausdorff shift-continuous topology on the semigroup B_ω^F is discrete and if a Hausdorff semitopological semigroup S contains B_ω^F as a proper dense subsemigroup then SB_ω^F is an ideal of S. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on B_ω^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 B_ω^F with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup B_ω^F 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 B_ω^F . , number=1, journal=Matematychni Studii, author=Gutik, O. V. and Mykhalenych, M. S., year=2023, month=Mar., pages=20-28