Korean J. Math.  Vol 22, No 1 (2014)  pp.37-43
DOI: https://doi.org/10.11568/kjm.2014.22.1.37

$\sigma$-complete Boolean algebras and basically disconnected covers

Chang Il Kim, Chang Hyeob Shin

Abstract


In this paper, we show that for any $\sigma$-complete Boolean subalgebra $\mathcal M$ of $\mathcal{R}(X)$ containing $Z(X)^\#$, the Stone-space $S(\mathcal M)$ of $\mathcal M$ is a basically diconnected cover of $\beta X$ and that the subspace $\{\alpha \mid \alpha$ is a fixed $\mathcal M$-ultrafilter$\}$ of the Stone-space $S(\mathcal M)$ is the the minimal basically disconnected cover of $X$ if and only if it is a basically disconnected space and $\mathcal M\subseteq \{\Lambda_X(A) \mid A \in Z(\Lambda X)^\#\}$.


Subject classification

54G05, 54G99, 54C01

Sponsor(s)



Full Text:

PDF

References


J. Ad ́amek, H. Herrilich, and G. E. Strecker, Abstract and concrete categories, John Wiley and Sons Inc. New York (1990). (Google Scholar)

L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Prince- ton, New York, (1960). (Google Scholar)

M. Henriksen, J. Vermeer, and R. G. Woods, Quasi-F-covers of Tychonoff spaces, Trans. Amer. Math. Soc. 303 (1987), 779–804. (Google Scholar)

Hager A. W. and Martinez J., C-epic compactifications, Topol. and its Appl. 117 (2002), 113–138. (Google Scholar)

M. Henriksen, J. Vermeer, and R. G. Woods, Wallman covers of compact spaces, Dissertationes Math. 283 (1989), 5–31. (Google Scholar)

S. Iliadis, Absolute of Hausdorff spaces, Sov. Math. Dokl. 4 (1963), 295-298. (Google Scholar)

C. I. Kim, Minimal covers and filter spaces, Topol. and its Appl. 72 (1996), 31–37. (Google Scholar)

J. Porter and R. G. Woods, Extensions and Absolutes of Hausdorff Spaces, (Google Scholar)

Springer, Berlin, (1988). (Google Scholar)

J. Vermeer, The smallest basically disconnected preimage of a space, Topol. Appl. (Google Scholar)

(1984), 217–232. (Google Scholar)


Refbacks

  • There are currently no refbacks.


ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr