Modified Whyburn semigroups (Q1102380)

Let \((X,m_1)\) and \((Y,m_2)\) be Hausdorff semigroups and \(f: X\to Y\) be a continuous semigroup homomorphism. Denote by \(W=X\dot \cup Y\) the disjoint union of \(X\) and \(Y\). Define the following topology on \(W\): \(V\) is open in \(W\) iff \(V\cap X\) and \(V\cap Y\) are open in \(X\), \(Y\) respectively, and \(f^{-1}(y)-V\) is compact for each \(y\) in \(V\cap Y\). This topology is called the modified Whyburn topology because it is finer than the topology defined by \textit{G. T. Whyburn} [Trans. Am. Math. Soc. 74, 344--350 (1953; Zbl 0053.12303)]. It is clear that the modified Whyburn topology is coarser than the disjoint union topology of \(X\) and \(Y\). However, under the modified Whyburn topology, a continuous multiplication is not easy to define on \(W\). This article is to give conditions which will ensure that \(W\) is a Hausdorff semigroup when the modified Whyburn topology is placed on \(W\). Moreover, if \(X\) is connected and \(f^{-1}(y)\) is not compact for each \(y\in Y\), then the authors prove that the semigroup \(Y\) has the zero multiplication if the Hausdorff semigroup \(W\) is assumed to be first countable.