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Characterizations of paracompactness and Lindelöfness by the separation property

Author: Yukinobu Yajima
Journal: Proc. Amer. Math. Soc. 131 (2003), 1297-1302
MSC (2000): Primary 54B10, 54D20
Published electronically: November 6, 2002
MathSciNet review: 1948123
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Abstract: The separation property in our title is that, for two spaces $X$ and $Y$, any two disjoint closed copies of $X$ in $Y$ are separated by open sets in $Y$. It is proved that a Tychonoff space $X$ is paracompact if and only if this separation property holds for the space $X$ and every Tychonoff space $Y$ which is a perfect image of $X\times \beta X$ (where $\beta X$ denotes the Stone-Cech compactification of $X$). Moreover, we give a characterization of Lindelöfness in a similar way under the assumption of paracompactness.

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Additional Information

Yukinobu Yajima
Affiliation: Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan

Keywords: Paracompact, separated by disjoint open sets, completely separated, Lindel\"{o}f
Received by editor(s): May 30, 2000
Received by editor(s) in revised form: November 12, 2001
Published electronically: November 6, 2002
Communicated by: Alan Dow
Article copyright: © Copyright 2002 American Mathematical Society

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