Stone-Čech remainders which make continuous images normal

Abstract:

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 $bf:Z \to Y$ 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.