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 | References | Similar Articles | Additional Information

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*.

- R. Engelking and S. Mrówka,
*On $E$-compact spaces*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.**6**(1958), 429–436. MR**0097042** - Zdeněk Frolík,
*A generalization of realcompact spaces*, Czechoslovak Math. J.**13(88)**(1963), 127–138 (English, with Russian summary). MR**155289** - Leonard Gillman and Meyer Jerison,
*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199** - K. D. Magill Jr. and J. A. Glasenapp,
*0-dimensional compactifications and Boolean rings*, J. Austral. Math. Soc.**8**(1968), 755–765. MR**0262130** - S. Mrówka,
*On $E$-compact spaces. II*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**14**(1966), 597–605 (English, with Russian summary). MR**206896** - S. Mrówka,
*Further results on $E$-compact spaces. I*, Acta Math.**120**(1968), 161–185. MR**226576**, DOI https://doi.org/10.1007/BF02394609

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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