• Korean Journal of Mathematics
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• v.16 no.4
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• pp.495-498
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• 2008

Every totally disconnected space is hereditarily disconnected. In this note, we provide an example of a hereditarily disconnected which is not a totally disconnected space. We further provide an example that not homogeneous space is the product of a totally disconnected and a connected.

• 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 $X\;{\times}\;Y$.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.1009-1018
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• 1997

Observing that for any $\beta_c$-Wallman functor $A$ and any Tychonoff space X, there is a cover $(C_1(A(X), X), c_1)$ of X such that X is $A$-disconnected if and only if $c_1 : C_1(A(X), X) \longrightarrow X$ is a homeomorphism, we show that every Tychonoff space has the minimal $A$-disconnected cover. We also show that if X is weakly Lindelof or locally compact zero-dimensional space, then the minimal G-disconnected (equivalently, cloz)-cover is given by the space $C_1(A(X), X)$ which is a dense subspace of $E_cc(\betaX)$.

• 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.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.

• 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.30 no.4
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• pp.357-375
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• 2023

In this paper, we introduce the concept of bipolar soft supra topological space and provide a characterization of the related concepts of bipolar soft supra closure and bipolar soft supra interior. We also establish a connection between bipolar soft supra topology and bipolar soft topology. Additionally, we present the concept of bipolar soft supra continuous mapping and examine the concept of bipolar soft supra compact topological space. A related result concerning the image of the bipolar soft supra compact space is proved. Finally, we identify the concepts of disconnected (connected) and strongly disconnected (strongly connected) space and derive several results linking them together. Relationships among these concepts are clarified with the aid of examples.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.2
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• pp.175-182
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• 2008

This paper is a revised version of [5]. In [5], we define' nearness between two points' in a topological space in many ways and show that a continuous function preserves one-sided nearness. We also show that a $T_1$-space is characterized by one-sided nearness exactly. In this paper, we introduce extremally disconnected spaces and show that the new topology generated by the set of equivalence classes as a base is extremally disconnected.

• 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 $\mathcal{M}{\subseteq}\{\Lambda_X(A){\mid}A{\in}Z({\Lambda}X)^{\sharp}\}$.