• 한국수학교육학회지시리즈B:순수및응용수학
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• 제23권4호
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• pp.329-337
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• 2016

In this paper, we show that every compactification, which is a quasi-F space, of a space X is a Wallman compactification and that for any compactification K of the space X, the minimal quasi-F cover QFK of K is also a Wallman compactification of the inverse image ${\Phi}_K^{-1}(X)$ of the space X under the covering map ${\Phi}_K:QFK{\rightarrow}K$. Using these, we show that for any space X, ${\beta}QFX=QF{\beta}{\upsilon}X$ and that a realcompact space X is a projective object in the category $Rcomp_{\sharp}$ of all realcompact spaces and their $z^{\sharp}$-irreducible maps if and only if X is a quasi-F space.

• East Asian mathematical journal
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• 제4권
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• pp.15-24
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• 1988

• 대한수학회논문집
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• 제17권4호
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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).

• 한국수학교육학회지시리즈A:수학교육
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• 제16권2호
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• pp.22-26
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• 1978

($\alpha$$_{x}$, $\alpha$X)와 ($\alpha$$_{Y}$ , $\alpha$Y)를 T$_2$ 공간 X와 Y의 Alexandroff base Compactification이라 할 때 $\alpha$fㆍ$\alpha$$_{x}$=$\alpha$$_{Y}$ f를 만족하는 open이고 연속인 함수 $\alpha$f:$\alpha$X$\longrightarrow$$\alpha$Y가 존재하는 연속함수 f:X$\longrightarrow$Y는 유일한 $\alpha$-extension $\alpha$f를 가지며 Category ABC를 T$_2$ 공간과 위와 같은 연속함수 f들의 Category라고 할 때 open이고 연속인 함수와 Compact space들의 Category는 Category ABC의 epireflective subcategory임을 밝혔다.

• 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.