We show that every Hausdorff Baire topology $\tau$ on $\mathcal{ C}=\langle a,b\mid a^2b=a, ab^2=b\rangle$ 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.
Categories: math.GR math.GN