• East Asian mathematical journal
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• v.30 no.1
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• pp.63-67
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• 2014

Observing that for any P'-space X, ${\Lambda}vX$ is a P'-space if vX is a weakly Lindel$\ddot{o}$f space, (${\Lambda}vX{\times}{\Lambda}Y$, ${\Lambda}_X{\times}{\Lambda}_Y$) is the minimal basically disconnected cover of $X{\times}Y$ for a countably locally weakly Lindel$\ddot{o}$f space Y.

• The Pure and Applied Mathematics
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• v.19 no.3
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• pp.273-279
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• 2012

In this paper, we construct an extension ($kX$, $k_X$) of a space X such that $kX$ is a weakly Lindel$\ddot{o}$ff space and for any continuous map $f:X{\rightarrow}Y$, there is a continuous map $g:kX{\rightarrow}kY$ such that $g|x=f$. Moreover, we show that ${\upsilon}X$ is Lindel$\ddot{o}$ff if and only if $kX={\upsilon}X$ and that for any P'-space X which is weakly Lindel$\ddot{o}$ff, $kX={\upsilon}X$.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.427-433
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• 2013

Observing that every Tychonoff space X has an extension $kX$ which is a weakly Lindel$\ddot{o}$f space and the minimal quasi-F cover $QF(kX)$ of $kX$ is a weakly Lindel$\ddot{o}$f, we show that ${\Phi}_{kX}:QF(kX){\rightarrow}kX$ is a $z^{\sharp}$-irreducible map and that $QF({\beta}X)=QF(kX)$. Using these, we prove that $QF(kX)=kQF(X)$ if and only if ${\Phi}^k_X:kQF(X){\rightarrow}kX$ is an onto map and ${\beta}QF(X)=(QF{\beta}X)$.

• East Asian mathematical journal
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• v.29 no.1
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• pp.83-89
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• 2013

Observing that every Tychonoff space X has a weakly Lindel$\ddot{o}$f extension ${\kappa}X$ and the minimal basically diconneted cover ${\Lambda}{\kappa}X$ of ${\kappa}X$ is weakly Lindel$\ddot{o}$f, we first show that ${\Lambda}_{{\kappa}X}:{\Lambda}{\kappa}X{\rightarrow}{\kappa}X$ is a $z^{\sharp}$-irreducible map and that ${\Lambda}{\beta}X={\beta}{\Lambda}{\kappa}X$. And we show that ${\kappa}{\Lambda}X={\Lambda}{\kappa}X$ if and only if ${\Lambda}^{\kappa}_X:{\kappa}{\Lambda}X{\rightarrow}{\kappa}X$ is an onto map and ${\beta}{\Lambda}X={\Lambda}{\beta}X$.

• Honam Mathematical Journal
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• v.35 no.2
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• pp.303-310
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• 2013

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 ${\beta}E_{cc}(X)=E_{cc}({\beta}X)$.

• Korean Journal of Mathematics
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• v.10 no.1
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• pp.45-51
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• 2002

Observing that a realcompactification Y of a space X is Wallman if and only if for any non-empty zero-set Z in Y, $Z{\cap}Y{\neq}{\emptyset}$, we will show that for any pseudo-Lindel$\ddot{o}$f space X, the minimal quasi-F $QF({\upsilon}X)$ of ${\upsilon}X$ is Wallman and that if X is weakly Lindel$\ddot{o}$, then $QF({\upsilon}X)={\upsilon}QF(X)$.

• Communications of the Korean Mathematical Society
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• v.21 no.2
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• pp.347-353
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• 2006

In this paper, we show that if the minimal basically disconnected cover ${\wedge}X_\imath\;of\;X_\imath$ is given by the space of fixed a $Z(X)^#$-ultrafilters on $X_\imath\;(\imath=1,2)\;and\;{\wedge}X_1\;{\times}\;{\wedge}X_2$ is a basically disconnected space, then ${\wedge}X_1\;{\times}\;{\wedge}X_2$ is the minimal basically disconnected cover of $X_1\;{\times}\;X_2$. Moreover, observing that the product space of a P-space and a countably locally weakly Lindelof basically disconnected space is basically disconnected, we show that if X is a weakly Lindelof almost P-space and Y is a countably locally weakly Lindelof space, then (${\wedge}X\;{\times}\;{\wedge}Y,\;{\wedge}_X\;{\times}\;{\wedge}_Y$) is the minimal basically disconnected cover of $X\;{\times}\;Y$.

• The Pure and Applied Mathematics
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• v.21 no.2
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• pp.113-120
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• 2014

In this paper, we first show that for any space X, there is a ${\sigma}$-complete Boolean subalgebra of $\mathcal{R}$(X) and that the subspace {${\alpha}{\mid}{\alpha}$ is a fixed ${\sigma}Z(X)^{\sharp}$-ultrafilter} of the Stone-space $S(Z({\Lambda}_X)^{\sharp})$ is the minimal basically disconnected cover of X. Using this, we will show that for any countably locally weakly Lindel$\ddot{o}$f space X, the set {$M{\mid}M$ is a ${\sigma}$-complete Boolean subalgebra of $\mathcal{R}$(X) containing $Z(X)^{\sharp}$ and $s_M^{-1}(X)$ is basically disconnected}, when partially ordered by inclusion, becomes a complete lattice.

• Honam Mathematical Journal
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• v.36 no.2
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• pp.253-261
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• 2014

In this paper, we first will show that for any space X and any Wallman sublattice $\mathcal{A}$ of $\mathcal{R}(X)$ with $Z(X)^{\sharp}{\subseteq}\mathcal{A}$, (${\Phi}^{-1}_{\mathcal{A}}(X)$, ${\Phi}_{\mathcal{A}}$) is the minimal quasi-F cover of X if and only if (${\Phi}^{-1}_{\mathcal{A}}(X)$, ${\Phi}_{\mathcal{A}}$) is a quasi-F cover of X and $\mathcal{A}{\subseteq}\mathcal{Q}_X$. Using this, if X is a locally weakly Lindel$\ddot{o}$f space, the set {$\mathcal{A}|\mathcal{A}$ is a Wallman sublattice of $\mathcal{R}(X)$ with $Z(X)^{\sharp}{\subseteq}\mathcal{A}$ and ${\Phi}^{-1}_{\mathcal{A}}(X)$ is the minimal quasi-F cover of X}, when partially ordered by inclusion, has the minimal element $Z(X)^{\sharp}$ and the maximal element $\mathcal{Q}_X$. Finally, we will show that any Wallman sublattice $\mathcal{A}$ of $\mathcal{R}(X)$ with $Z(X)^{\sharp}{\subseteq}\mathcal{A}{\subseteq}\mathcal{Q}_X$, ${\Phi}_{\mathcal{A}_X}:{\Phi}^{-1}_{\mathcal{A}}(X){\rightarrow}X$ is $z^{\sharp}$-irreducible if and only if $\mathcal{A}=\mathcal{Q}_X$.