Про топологізацію піднапівгруп біциклічного моноїда

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) Hausdorff Baire topology on the upper subsemigroup \(\mathcal{C}_+(a,b)\) (down subsemigroup \(\mathcal{C}_-(a,b)\) of \({\mathcal{C}}(p,q)\) is discrete and the same statement holds for the bicyclic monoid.