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We prove that the semigroup operation of a topological semigroup extends to a continuous semigroup operation on its Stone-Čech compactification provided is a pseudocompact openly factorizable space, which means that each map to a second countable space can be written as the composition of an open map onto a second countable space and a map . We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.
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