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We show that in the class of Lindel\"of {C}ech-complete spaces the property of being C-embedded is quite well-behaved. It admits a useful characterization that can be used to show that products and perfect preimages of C-embedded spaces are again C-embedded. We also show that both properties, Lindel\"of and {C}ech-complete, are needed in the product result.
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Dow et al. (2024) studied this question.
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