On topologization of subsemigroups of the bicyclic monoid

Abstract: We show that if a subsemigroup $S$ of the bicyclic monoid ${\mathscr{C}}(p,q)$ contains infinitely many idempotents then $S$ admits only the discrete Hausdorff shift-continuous topology. Also we proof that every right-continuous (left-continuous\emph) Hausdorff Baire topology on the semigroup $\mathscr{C}+(a,b)$ $(\mathscr{C}-(a,b))$ is discrete and the same statement holds for the bicyclic monoid.