• 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 $X\;{\times}\;Y$.

• Korean Journal of Mathematics
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• v.22 no.1
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• pp.37-43
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• 2014

In this paper, we show that for any ${\sigma}$-complete Boolean subalgebra $\mathcal{M}$ of $\mathcal{R}(X)$ containing $Z(X)^{\sharp}$, 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 $\mathcal{M}{\subseteq}\{\Lambda_X(A){\mid}A{\in}Z({\Lambda}X)^{\sharp}\}$.

• The Pure and Applied Mathematics
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• v.17 no.2
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• pp.167-173
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• 2010

In this paper, we introduce the concept of a weakly P-space which is a generalization of a P-space and prove that for any covering map f : $X{\rightarrow}Y$, X is a weakly P-space if and only if Y is a weakly P-space. Using these, we investigate the minimal basically disconnected cover of weakly P-spaces and their products.

• Journal for History of Mathematics
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• v.16 no.1
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• pp.73-78
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• 2003

Observing that if f: $Y{\leftrightarro}$Χ is a covering map and Χ is a countably locally weakly Lindelof space, then Y is countably locally weakly Lindelof and that every dense countably weakly Lindelof subspace of a basically disconnected space is basically disconnected, we show that for a countably weakly Lindelof space Χ, its minimal basically disconnected cover ${\bigwedge}$Χ is given by the filter space of fixed ${\sigma}Ζ(Χ)^#$- ultrafilters.

• 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.

• 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.

• Journal for History of Mathematics
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• v.15 no.2
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• pp.161-168
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• 2002

We show that if the Stone-Cech compactification of $\textit{AX}$ and the minimal basically disconnected cove. of $\beta$Χ we homeomorphic and every real $\sigma$$Z(X)^#$-ultrafilter on X has the countable intersection property, then there is a covering map from $\nu$(ΛΧ) to $\nu$Χ and every real $\sigma$$Z(X)^#$-ultrafilter on Χ has the countable intersection property if and only if there is a homeomorphism from the Hewitt realcompactification of ΛΧ to the minimal basically disconnected space of $\nu$Χ.

• Journal for History of Mathematics
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• v.11 no.2
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• pp.83-90
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• 1998

Observing that for any compact space X, the minimal basically disconnected cover ${\bigwedge}Λ_X$ : ${\bigwedge}Λ_X{\leftrightarro}$ is ${\sigma}Z^#$-irreducible, we will show that the projective objects in the category of compact spaces and ${\sigma}Z^#$-irreducible maps are precisely basically disconnected spaces.

• The Pure and Applied Mathematics
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• v.9 no.1
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• pp.9-17
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• 2002

In this Paper, we will show that every basically disconnected space is a projective object in the category $Tych_{\sigma}$ of Tychonoff spaces and $_{\sigma}Z^{#}$ -irreducible maps and that if X is a space such that ${\Beta} {\Lambda} X={\Lambda} {\Beta} X$, then X has a projective cover in $Tych_{\sigma}$. Moreover, observing that for any weakly Linde1of space, ${\Lambda} X : {\Lambda} X\;{\longrightarrow}\;X$ is $_{\sigma}Z^{#}$-irreducible, we will show that the projective objects in $wLind_{\sigma}$/ of weakly Lindelof spaces and $_{\sigma}Z^{#}$-irreducible maps are precisely the basically disconnected spaces.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.709-718
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• 2002

Observing that each Tychonoff space X has the minimal basically disconnected cover (ΛX, Λ$\sub$X/) and the .realcompact-ification $\upsilon$X, we introduce a concept of stable $\sigma$Z(X)#-ultrafilters and give internal characterizations of Tychonoff spaces X for which Λ($\upsilon$X) : $\upsilon$(ΛX).