Stone-Cech Remainders which Make Continuous Images Normal

• Published in: Proceedings of the American Mathematical Society 1989, Vol.106 (3), p.839-842
• Main Authors: Fleissner, William, Levy, Ronnie
• Format: Article
• Language: English
• Subjects: College mathematics; Compactification; First countable; Hausdorff spaces; Normal spaces; Regular spaces; Remainders
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-1989-0963571-8

If $f$ is a continuous surjection from a normal space $X$ onto a regular space $Y$, then there are a space $Z$ and a perfect map $bf: Z \rightarrow Y$ extending $f$ such that $X \subset Z \subset \beta X$. If $f$ is a continuous surjection from normal $X$ onto Tychonov $Y$ and $\beta X\backslash X$ is sequential, then $Y$ is normal. More generally, if $f$ is a continuous surjection from normal $X$ onto regular $Y$ and $\beta X\backslash X$ has the property that countably compact subsets are closed (this property is called $C$-closed), then $Y$ is normal. There is an example of a normal space $X$ such that $\beta X\backslash X$ is $C$-closed but not sequential. If $X$ is normal and $\beta X\backslash X$ is first countable, then $\beta X\backslash X$ is locally compact.