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# Stone-Cech Remainders which Make Continuous Images Normal

William Fleissner and Ronnie Levy
Proceedings of the American Mathematical Society
Vol. 106, No. 3 (Jul., 1989), pp. 839-842
DOI: 10.2307/2047443
Stable URL: http://www.jstor.org/stable/2047443
Page Count: 4
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.