On Two Problems of Mostert and Shields
J. H. Ursell · Proceedings of the American Mathematical Society · 1963
A topological semigroup S is a topological space S such that the set S has a binary composition which is associative and continuous with respect to the topology.In [2], P. S. Mostert and A. L. Shields gave a list of problems in topological semigroups.In this and subsequent notes we give partial answers to several of these problems and to others suggested to the author by R. P. Hunter.We give here the answers to problems P2 and P6 of [2], which are: P2.Let 5 be a locally-compact, connected, Hausdorff topological semigroup with zero and identity.Is there is a compact, connected subsemigroup M containing the zero and identity?P6. Let S be a compact, connected, Hausdorff topological semigroup with identity and which is not a group.Does 5 admit a continuous, nontrivial homomorphism on to some (/)-semigroup?In each case we give a counter-example which answers these questions in the negative.The author wishes to state here his gratitude to his supervisor, M. F. Atiyah, for discussions, and to R. P. Hunter for communications by letter, which have enabled him to present both the theorems in this paper in a far more elegant manner than would otherwise have been the case.