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Grassmann 多樣體上의 Vector Bundles에 關한 硏究
金來善,金仁洙 전주대학교 1986 論文集 Vol.15 No.-
本 論文은 Grasemann 多樣體의 性質을 究明하기 위하여 Vector bundle 上에 Gaussmap을 利用하여 任意의 vector bundle과 Grassmann 多樣體 ?? (??)상의 vector bundle 사이에 bundle map을 주고 두 bundle map 사이에 homotopy를 정의하여 다음의 정리를 證明하였다. (定理) 두 vector bundle ξ와 ??(??)=( ?? )이 주어지고, 만일 ?? 이 ?? 에 의하여 誘導되어지면 j˚f.j˚f'은 bundle homotopic이다. 단 (f,g), (f',g')은 ξ에서 ??으로의 bundle map이다. The Grassmann manifolds are defind by Stiefel manifolds and the general idea of vector bundle theories can be found in [3], [4], [5]. In this paper, we shall discuss vector bundles being based on Grassmann manifolds which is grown by [3], [5]. The main theorem of this paper it that j˚f-j'˚f', where (f,g), (f',g') are bundle maps and ?? is the bundle map induced by the inclusion ?? . We have defined projective n-space ?? to be the set of all pairs {p, -p} for p∈??⊂??. We could also have define the Grassmann manifold ?? as follows: For a m-dimensional vector space ??, {v₁,…v??} which are linearly independent in ?? is called a n-frame of ??. Put ?? is the set of all n-frames of ?? and ?? is called the Stiefel manifold of ?? which has the relative topology as a subset of ?? (n-times). Let ?? is the set of all n-dimensional subspace of ??. Then there is the projection π: ?? by π(v₁,…??)=[v₁,…??], where [v₁,…??] is the n-dimensional subspace of ?? generated by {v₁, …??}. ?? has the quotient topology by π. Put ?? is the set of all orthogonormal n-frames of ??. For a n-frame (v₁,…??) of ??, there is a continuous function g : ?? such that g(v₁, …??)=(v₁',…??'). In fact v₁'=??, ……, ?? Let ?? , then by the following commutative diagram ?? We see that the topology of ?? is the quotient topology by ??. For example, V₁ ?? and G₁ ?? .
On Some Arcwise Connected Hyperspaces
Kim, Rae-Seon 전주대학교 자연과학종합연구소 2000 전주대학교 자연과학연구소 학술논문집 Vol.13 No.-
In this paper, we investigated the relationships between the space X and its hyperspaces concerning the properties of compactness, local connectedness connectedness and arcwise connectedness. Then we proved Theorem 2.1, Corollary 2.2, Theorem 2.3, and argued (1)[Theorem 2.4] If X is a Hausdorff space, then X is compact and connected if and only if 2^(x)(or C(X)) is compact arcwise connected Hausdorff space. (2)[Theorem 2.5] If X is a compact Hausdorff space, then the connectedness of X is equivalent to the arcwise connectedness of K(X) (or F_(n)(X), F(X), C_(k)(X)).
On the Convergent Seguences in C(X)
Kim, Rae-Seon,Choi, Young-Ho 전주대학교 교육문제연구소 1992 敎育論叢 Vol.7 No.-
For a convergent sequence {X_(n)}_(n) of subcontinua of a metric continuum X which converges to X_(0)∈C(X), the sequence {C(X_(n))}_(n) does not converge to C(X_(0)) in general. If {X_(n)}_(n) converges to X_(0) 0-regularly, then {C(X_(n))}_(n) converges to C(X_(0)) [2, pp.518-520]. In this paper, we give other sufficient condition for the convergence of {C(X_(n))}, namely h-regular convergence ,and we study on the relation between them. Let (X,d) be a metric continuum, i.e. compact connected metric space consisting of more than one point. Let 2^X = {A⊂X : A is nonempty and closed} and let C(X) = {A∈2^(X) : A is connected}. For each A∈2^(X) and ε>0, let N(ε,A) = {x∈X : d(x,a)<ε for some a∈A}. If A,B∈2^(X), let H(A,B) = inf{ε>0 : A⊂N{ε,B} B⊂N(ε,A)} ; we call H the Hausdorff metric for 2^(X)(or C(X)) induced by d. The spaces 2^(X) and C(X), with the Hausdorff metric, are called hyperspaces of X. We have the following notations; if x∈M⊂X, let T(x,M)={A∈C(M):x∈A}, C(M)={A∈C(X):A⊂M}, and M^(*)={{x}:x∈M}. Let {A_(n)}_(n) be a sequence in 2^(X). Let L_(i)A_(n) be the set of all x∈X such that if U is a neighborhood of x, then U ∩ A_(n) ≠ φ for all but finitely many n ; L_(s)A_(n) be the set of all x∈X such that if U is a neighborhood of x, then U ∩ A_(n) ≠ φ for infinitely many n. If L_(i)A_(n) = L_(s)A_(n) = A, then we say that the sequence {A_(n)}_(n) conerges to A, written L_(t)A_(n) = A or A_(n) → A. It is known that if {A_(n)}_(n) is a sequence in 2^(X)(or C(X)), then A_(n) → A if and only if H(A_(n),A) → 0. ([2])