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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A characterization of $N$-compact spaces


Author: Kim-peu Chew
Journal: Proc. Amer. Math. Soc. 26 (1970), 679-682
MSC: Primary 54.53
DOI: https://doi.org/10.1090/S0002-9939-1970-0267534-5
MathSciNet review: 0267534
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Abstract: In this paper, we prove the following theorem: Theorem A. A $0$-dimensional space $X$ is $N$-compact if and only if every clopen ultrafilter on $X$ with the countable intersection property is fixed, where $N$ is the space of all natural numbers. Two consequences of Theorem A are as follows: Theorem B. Suppose that $X$ and $Y$ are $N$-compact spaces. A mapping $\phi$ from the Boolean ring $\mathfrak {B}(X)$ of all clopen subsets of $X$ onto the Boolean ring $\mathfrak {B}(X)$ of all clopen subsets of $Y$ is an isomorphism with the property that $\bigcap \nolimits _{i = 1}^\infty {{A_i} = \emptyset ({A_i} \in \mathfrak {B}(X))}$ implies $\bigcap \nolimits _{i = 1}^\infty \phi ({A_i}) = \emptyset$ if and only if there exists a homeomorphism $h$ from $X$ onto $Y$ such that $\phi (A) = h[A]$ for each $A$ in $\mathfrak {B}(X)$. Theorem C. A $0$-dimensional space $X$ is $N$-compact if and only if the collection of all the countable clopen coverings of $X$ is complete.


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Keywords: <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$E$">-completely regular, <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$E$">-compact, <IMG WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$N$">-compact, clopen ultrafilter, countable intersection property, Boolean ring
Article copyright: © Copyright 1970 American Mathematical Society