• 충청수학회지
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• 제22권4호
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• pp.831-841
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• 2009

For a map f : A $\rightarrow$ X, we define and study a concept of $G^f$-space for a map, which is a generalized one of a G-space. Any G-space is a $G^f$-space, but the converse does not hold. In fact, $S^2$ is a $G^{\eta}$-space, but not G-space. We show that X is a $G^f$-space if and only if $G_n$(A, f,X) = $\pi_n(X)$ for all n. It is clear that any $H^f$-space is a $G^f$-space and any $G^f$-space is a $W^f$-space. We can also obtain some results about $G^f$-spaces in Postnikov systems for spaces, which are generalization of Haslam's results about G-spaces.

• 충청수학회지
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• 제27권4호
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• pp.735-743
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• 2014

We obtain a splitting theorem which characterizes when a given space is a catesian product of an $H^f$-space, and also obtain a dual theorem for a co-$H^g$-space. Then we get Dula and Gottlieb's results as corollaries.

• 충청수학회지
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• 제20권3호
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• pp.245-259
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• 2007

We define and study a concept of $T^f$-space for a map, which is a generalized one of a T-space, in terms of the Gottlieb set for a map. We show that X is a $T_f$-space if and only if $G({\Sigma}B;A,f,X)=[{\Sigma}B,X]$ for any space B. For a principal fibration $E_k{\rightarrow}X$ induced by $k:X{\rightarrow}X^{\prime}$ from ${\epsilon}:PX^{\prime}{\rightarrow}X^{\prime}$, we obtain a sufficient condition to having a lifting $T^{\bar{f}}$-structure on $E_k$ of a $T^f$-structure on X. Also, we define and study a concept of co-$T^g$-space for a map, which is a dual one of $T^f$-space for a map. We obtain a dual result for a principal cofibration $i_r:X{\rightarrow}C_r$ induced by $r:X^{\prime}{\rightarrow}X$ from ${\iota}:X^{\prime}{\rightarrow}cX^{\prime}$.

• 충청수학회지
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• 제28권4호
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• pp.603-614
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• 2015

For a map $p:X{\rightarrow}A$, we define and study a concept of $G^{\prime}_p$-space for a map, which is a generalized one of a G'-space. Any G'-space is a $G^{\prime}_p$-space, but the converse does not hold. In fact, $CP^2$ is a $G^{\prime}_{\delta}$-space, but not a G'-space. It is shown that X is a $G^{\prime}_p$-space if and only if $G^n(X,p,A)=H^n(X)$ for all n. We also obtain some results about $G^{\prime}_p$-spaces and homology decompositions for spaces. As a corollary, we can obtain a dual result of Haslam's result about G-spaces and Postnikov systems.

• 대한수학회논문집
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• 제37권1호
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• pp.105-112
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• 2022

Let g : X ⟶ Y and f : Y ⟶ Z be two maps between real normed linear spaces. Then f is called generalized isometry or g-isometry if for each x, y ∈ X, ║f(g(x)) - f(g(y))║ = ║g(x) - g(y)║. In this paper, under special hypotheses, we prove that each generalized isometry is affine. Some examples of generalized isometry are given as well.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제14권4호
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• pp.289-306
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• 2007

We define and study a concept of $H^f-space$ for a map, which is a generalized concept of an H-space, in terms of the Gottlieb set for a map. For a principal fibration $E_{\kappa}{\rightarrow}X$ induced by ${\kappa}:X{\rightarrow}X'\;from\;{\epsilon}:\;PX'{\rightarrow}X'$, we can obtain a sufficient condition to having an $H^{\bar{f}}-structure\;on\;E_{\kappa}$, which is a generalization of Stasheff's result [17]. Also, we define and study a concept of $co-H^g-space$ for a map, which is a dual concept of $H^f-space$ for a map. Also, we get a dual result which is a generalization of Hilton, Mislin and Roitberg's result [6].

• 대한수학회논문집
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• 제13권1호
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• pp.109-121
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• 1998

Let X,Y be connected, locally connected, semilocally simply connected and $f,g : X \to Y$ be a pair of maps. We find an upper bound of the Reidemeister number R(f,g) by using the regular coverig spaces.

• 충청수학회지
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• 제9권1호
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• pp.77-87
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• 1996

For any map $v:X{\rightarrow}Y$, the generalized Gottlieb set $G({\Sigma}A;X,v,Y)$ with respect to v is a subgroup of $[{\Sigma}A,Y]$. If $v:X{\rightarrow}Y$ has a left homotopy inverse $u:X{\rightarrow}Y$, then for any $f{\in}G({\Sigma}A;X,v,Y)$, $g{\in}G({\Sigma}A;X,u,Y)$, the function spaces $L({\Sigma}A,X;uf)$ and $L({\Sigma}A,X;g)$ have the same homotopy type.

• 대한수학회논문집
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• 제12권3호
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• pp.701-707
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• 1997

In [8], we introduce the relative root Nielsen number N(f;X, A, c) for maps of pairs of spaces $f : (X, A) \to (Y, B)$. From it, we obtain some immediate consequences of the definition and illustrate it by some examples. We consider the question whether there exists a map $g : (X, A) \to (Y, B)$ homotopic to a given map $f : (X, A) \to (Y, B)$ which has precisely N(f;X, A, c) roots, that is, the minimum theorem for N(f;X, A, c).

• 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.