Openly Factorizable Spaces and Compact Extensions of Topological Semigroups

We prove that the semigroup operation of a topological semigroup S extends to a continuous semigroup operation on its the Stone-Čech compactification βS provided S is a pseudocompact openly factorizable space, which means that each map f : S → Y to a second countable space Y can be written as the composition f = g ◦ p of an open map p : X → Z onto a second countable space Z and a map g : Z → Y . We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces. This paper was motivated by the problem of detecting topological semigroups that embed into compact topological semigroups. One of the ways to attack this problem is to find conditions on a topological semigroup S guaranteeing that the semigroup operation of S extends to a continuous semigroup operation on the Stone-Čech compactification βS of S. A crucial step in this direction was made by E.Reznichenko [15] who proved that the semigroup operation on a pseudocompact topological semigroup S extends to a separately continuous semigroup operation on βS. In this paper we show that the extended operation on βS is continuous if the space S is separable and openly factorizable, which means that each continuous map f : S → Y to a second countable space Y can be written as the composition f = g ◦ p of an open continuous map p : X → Z onto a second countable space Z and a continuous map g : Z → Y . The class of openly factorizable spaces turned to be interesting by its own so we devote Sections 2, 3 to studying such spaces. We recall that the Stone-Čech compactification of a Tychonov space X is a compact Hausdorff space βX containing X as a dense subspace so that each continuous map f : X → Y to a compact Hausdorff space Y extends to a continuous map f̄ : βX → Y . Replacing in this definition compact Hausdorff spaces by real complete spaces we obtain the definition of the Hewitt completion υX ofX . We recall that a topological space X is real complete if X is homeomorphic to a closed subspace of some power R κ of the real line. Thus a Hewitt completion of a Tychonov space X is a real complete space υX containing X as a dense subspace so that each continuous map f : X → Y to a real complete space Y extends to a continuous map υf : υX → Y . By [6, 3.11.16], the Hewitt completion υX can be identified with the subspace {x ∈ βX : G ∩X 6= ∅ for any Gδ-set G ⊂ βX with x ∈ G} of the Stone-Čech compactification βX of X . 1991 Mathematics Subject Classification. 22A15; 54B30; 54C20; 54C08; 54D35.