Geodesic
Openly factorizable spaces and compact extensions of topological semigroups
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 1, pp. 113-131.

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We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its Stone-Čech compactification $\beta S$ 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.
Classification : 22A15, 54B30, 54C08, 54C20, 54D35
Keywords: topological semigroup; semigroup compactification; inverse spectrum; pseudocompact space; openly factorizable space; openly generated space; Eberlein compact; Corson compact; Valdivia compact 
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Banakh, Taras; Dimitrova, Svetlana. Openly factorizable spaces and compact extensions of topological semigroups. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 1, pp. 113-131. http://geodesic.mathdoc.fr/item/CMUC_2010__51_1_a9/