• 충청수학회지
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• 제26권2호
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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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• 제29권1호
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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 ${\beta}{\Lambda}X={\Lambda}{\beta}X$.

• Kyungpook Mathematical Journal
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• 제46권4호
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• pp.503-507
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• 2006

Let X be a Tychonoff space and ${\sum}(X)$ the set of all the subrings of C(X) that contain $C^*(X)$. For any A(X) in ${\sum}(X)$ suppose $_{{\upsilon}A}X$ is the largest subspace of ${\beta}X$ containing X to which each function in A(X) can be extended continuously. Let us write A(X) ~ B(X) if and only if $_{{\upsilon}A}X=_{{\upsilon}B}X$, thereby defining an equivalence relation on ${\sum}(X)$. We have shown that an A(X) in ${\sum}(X)$ is isomorphic to C(Y ) for some space Y if and only if A(X) is the largest member of its equivalence class if and only if there exists a subspace T of ${\beta}X$ with the property that A(X)={$f{\in}C(X):f^*(p)$ is real for each $p$ in T}, $f^*$ being the unique continuous extension of $f$ in C(X) from ${\beta}X$ to $\mathbb{R}^*$, the one point compactification of $\mathbb{R}$. As a consequence it follows that if X is a realcompact space in which every $C^*$-embedded subset is closed, then C(X) is never isomorphic to any A(X) in ${\sum}(X)$ without being equal to it.