$C$-embedding, Lindelöfness, and Čech-completeness

We show that in the class of Lindelöf Čech-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, Lindelöf and Čech-complete, are needed in the product result.