$C$-embedding, Lindel\"ofness, and \v{C}ech-completeness

TL;DR

This paper investigates the properties of $C$-embedded spaces within Lindel"of e9ech-complete spaces, providing characterizations and demonstrating stability under products and preimages, emphasizing the necessity of both Lindel"of and e9ech-completeness conditions.

Contribution

It offers a new characterization of $C$-embedded spaces in Lindel"of e9ech-complete spaces and proves their stability under products and perfect preimages.

Findings

$C$-embeddedness is well-behaved in Lindel"of e9ech-complete spaces.

Products and perfect preimages of $C$-embedded spaces remain $C$-embedded.

Both Lindel"of and e9ech-completeness are essential for these properties.

Abstract

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.