On topologization of the bicyclic monoid

We construct two non-discrete inverse semigroup T₁-topologies and a compact inverse shift-continuous T₁-topology on the bicyclic monoid C(p,q). Also we give conditions on a T₁-topology τ on C(p,q) to be discrete. In particular, we show that if τ is an inverse semigroup T₁-topology on C(p,q) which satisfies one of the following conditions: τ is Baire, τ is quasi-regular or τ is semiregular, then τ is discrete.