On some generalization of the bicyclic semigroup: the topological version

We show that every Hausdorff Baire topology $\tau$ on $\mathcal{C}=⟨a,b\mid a^{2}b=a,a{b}^2=b⟩$ such that $(\mathcal{C},\tau )$ is a semitopological semigroup is discrete and we construct a nondiscrete Hausdorff semigroup topology on $\mathcal{C}$. We also discuss the closure of a semigroup $\mathcal{C}$ in a semitopological semigroup and prove that $\mathcal{C}$ does not embed into a topological semigroup with the countably compact square.