C. Eberhart and J. Selden showed that the only Hausdorff topology on the bicyclic semigroup which makes it a topological semigroup is the discrete topology. A related result proved in this paper is the following: Let \(W_{\alpha}\) be the \(\alpha\)-bisimple semigroup. The only locally compact Hausdorff semigroup topology on \(W_{\alpha}\) is discrete. Here \(\alpha\) is an ordinal and \(W_ 1\) as the bicyclic semigroup. \(W_{\alpha}\) is defined as follows: Let \(H_{\alpha}\) denote the set of all ordinals less than \(W_{\alpha}\). Let \(+\) denote usual ordinal addition. Let \(W_{\alpha}=H_{\alpha}\times H_{\alpha}\). Define a semigroup operation * on \(W_{\alpha}\) by \((\beta,\gamma)*(\delta,\eta)=(\beta +(Max\{\gamma,\delta \}-\gamma),\eta +(Max\{\gamma,\delta \}-\delta)),\) then \((W_{\alpha},*)\) is a bisimple inverse semigroup called \(\alpha\)-bicyclic semigroup.