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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0267534-5
Keywords: $ E$-completely regular, $ E$-compact, $ N$-compact, clopen ultrafilter, countable intersection property, Boolean ring
Article copyright: © Copyright 1970 American Mathematical Society