On topologization of subsemigroups of the bicyclic monoid

We show that if a subsemigroup $S$ of the bicyclic monoid $\mathcal{C}(p,q)$ 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 ${\mathcal{C}}_{+}(a,b)$ $({\mathcal{C}}_{-}(a,b))$ is discrete and the same statement holds for the bicyclic monoid.