  We show that in the class of Lindel\"of \v{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 \v{C}ech-complete, are needed in the product result.
