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Openly factorizable spaces and compact extensions of topological semigroups

Taras O. Banakh; Svetlana Dimitrova — 2010

Commentationes Mathematicae Universitatis Carolinae

We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its Stone-Čech compactification $β S$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f : S \to Y$ to a second countable space $Y$ can be written as the composition $f = g \circ p$ of an open map $p : X \to Z$ onto a second countable space $Z$ and a map $g : Z \to Y$ . We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.

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