On non-topologizable semigroups

Abstract

We find anti-isomorphic submonoids ${\mathcal{C}}_{+}(a,b)$ and ${\mathcal{C}}_{-}(a,b)$ of the bicyclic monoid $\mathcal{C}(a,b)$ with the following properties: every Hausdorff left-continuous (right-continuous) topology on ${\mathcal{C}}_{+}(a,b)$ (${\mathcal{C}}_{-}(a,b)$) is discrete and there exists a compact Hausdorff topological monoid $S$ which contains ${\mathcal{C}}_{+}(a,b)$ (${\mathcal{C}}_{-}(a,b)$) as a submonoid. Also, we construct a non-discrete right-continuous (left-continuous) topology $\ddot{I}{„}_p^{+}$ ($\ddot{I}{„}_p^{-}$) on the semigroup ${\mathcal{C}}_{+}(a,b)$ (${\mathcal{C}}_{-}(a,b)$) which is not left-continuous (right-continuous).

9 pages