# MINIMAL CLOZ-COVERS OF κX

Jo, Yun Dong;Kim, ChangIl

• Accepted : 2013.05.13
• Published : 2013.06.25
• 30 3

#### Abstract

In this paper, we first show that $z_{{\kappa}X}:E_{cc}({\kappa}X){\rightarrow}{\kappa}X$ is $z^{\sharp}$-irreducible and that if $\mathcal{G}(E_{cc}({\beta}X))$ is a base for closed sets in ${\beta}X$, then $E_{cc}({\kappa}X)$ is $C^*$-embedded in $E_{cc}({\beta}X)$, where ${\kappa}X$ is the extension of X such that $vX{\subseteq}{\kappa}X{\subseteq}{\beta}X$ and ${\kappa}X$ is weakly Lindel$\ddot{o}$f. Using these, we will show that if $\mathcal{G}({\beta}X)$ is a base for closed sets in ${\beta}X$ and for any weakly Lindel$\ddot{o}$f space Y with $X{\subseteq}Y{\subseteq}{\kappa}X$, ${\kappa}X=Y$, then $kE_{cc}(X)=E_{cc}({\kappa}X)$ if and only if ${\beta}E_{cc}(X)=E_{cc}({\beta}X)$.

#### Keywords

Stone-space;weakly Linel$\ddot{o}$f space;cloz-space;covering map

#### References

1. L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, New York, 1960.
2. M. Henriksen, J. Vermeer, and R. G.Woods, Quasi-F covers of Tychonoff spaces, Trans. Amer. Math. Soc. 303 (1987), 779-804.
3. M. Henriksen, J. Vermeer, and R. G. Woods, Wallman covers of compact spaces, Dissertationes Math. 283 (1989), 5-31.
4. M. Henriksen and R. G. Woods, Cozero complement spaces; When the space of minimal prime ideals of a C(X) is compact, Topology Appl. 141 (2004), 147-170. https://doi.org/10.1016/j.topol.2003.12.004
5. S. Iliadis, Absolute of Hausdorff spaces, Sov. Math. Dokl. 2 (1963), 295-298.
6. C. I. Kim, Cloz-covers of Tychonoff spaces, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 18 (2011), 361-386. https://doi.org/10.7468/jksmeb.2011.18.4.361
7. C. I. Kim, Minimal cloz-covers and Boolean Algebras, Korean J. Math. 20 (2012), 517-524. https://doi.org/10.11568/kjm.2012.20.4.517
8. Y. S. Yun and C. I. Kim, An extension which is a weakly Lindelof space, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 19 (2012), 273-279. https://doi.org/10.7468/jksmeb.2012.19.3.273
9. J. Porter and R. G. Woods, Extensions and Absolutes of Hausdorff Spaces, Springer, Berlin, 1988.