In this paper, by defining two new spectral sets, we give the necessary and sufficient conditions for Browder's theorem and Weyl's theorem for bounded linear operator T and f(T), where f∈H(σ(T)) and H(σ(T)) denotes the set of all analytic functions on an open neighborhood of σ(T).

Theorem 1 A topological space X is regular if and only if each point x in X and any open neighborhood U of x,there is a closed neighborhood V of x such that VU.

定理2 拓扑空间 X 为正规空间,当且仅当对于 X 中的任意不相交的闭集 A、B、A、B分别有不相交的闭邻域 U、V,使U∩V=.

Taking the Club at Legacy Garden, Karl Community and Fenghua Tiancheng as examples, this article further explains Chen's ideas on plain and plummily good architecture, which includes simple and chic form, open neighborhood, integrity of living behavior and space, and harmony of ecological environment and architecture, etc.

I describe Riemann surfaces of constant curvature -1 with the property that the length of its shortest simple closed geodesic is maximal with respect to an open neighborhood in the corresponding Teichmüller space.

For any[Figure not available: see fulltext.]>amp;gt;0 there exists an open neighborhood V of the set K such that any function[Figure not available: see fulltext.]-analytic on K coincides in some neighborhood of the set K with a function analytic in V.

With some mild assumptions, it is shown that there exists an open neighborhood around the minimizer so that our scheme applied to any point in the neighborhood will always give the correct optimal eigenvalue multiplicity.

Dirichlet solutions are valid in the open neighborhood of the hexagon, due to singular boundary conditions.

It is also shown that the restriction of a linear single-valued map to a convex set containing an open neighborhood of the origin is always limit transitive.

In this paper we prove four theorems,all part of a program to pronounce regular and normal space. Theorem 1 A topological space X is regular if and only if each point x in X and any open neighborhood U of x,there is a closed neighborhood V of x such that VU. Theorem 2 A topological space X is normal if and only if each closed set A in X and any open neighborhood U of A,there is a closed neighborhood V of A such that V U. Theorem 3 A topological space X is regular if and only if any point...

In this paper we prove four theorems,all part of a program to pronounce regular and normal space. Theorem 1 A topological space X is regular if and only if each point x in X and any open neighborhood U of x,there is a closed neighborhood V of x such that VU. Theorem 2 A topological space X is normal if and only if each closed set A in X and any open neighborhood U of A,there is a closed neighborhood V of A such that V U. Theorem 3 A topological space X is regular if and only if any point x in X and any closed sct B in X,x∈B,there are disjoint closed neighborhoods of x and of B. Theorem 4 A topological space X is normal if and only if any disjoint closcd sets A and B in X,there are closed neighorhoods U,V of A and of B,such that,U∩V=

本文给出了正则和正规空间的4个判定定理:定理1 拓扑空间 X 为正则空间,当且仅当对于 X 中的任一点x 以及 x 中不含 x 的任一闭集 B,x、B 分别有闭邻域 U、V,使得U∩V=.定理2 拓扑空间 X 为正规空间,当且仅当对于 X 中的任意不相交的闭集 A、B、A、B分别有不相交的闭邻域 U、V,使U∩V=.定理3 拓扑空间 X 为正则空间,当且仅当对 X 中的任一点 x 以及不含点 x 的任意闭集B,分别有 x,B 的闭邻域 U、V,使得 i(U)∩i(V)=.定理4 拓扑空间 X 为正规空间,当且仅当对 X 中的任意两个不相交的闭集 A、B,A、B 分别有闭邻域 U、V,使得 i(U)∩i(V)=.

In this paper we prove that several weak base metrization theorems, the main results are as follows: (1) A topology space X is metrization if and only if X has a weak base satisfied. If for each covergent seguence {x n}→x and each open neighborhood U of x there exists m∈N such that T(x,m)U and the femilly of all member of B that meet both T(x,m) and X U is finite, where T(x,m)={x n:n≥n}. (2) A topology space X is metrization if and only if X has a week development G 1,G 2,… Such that for each natural...

In this paper we prove that several weak base metrization theorems, the main results are as follows: (1) A topology space X is metrization if and only if X has a weak base satisfied. If for each covergent seguence {x n}→x and each open neighborhood U of x there exists m∈N such that T(x,m)U and the femilly of all member of B that meet both T(x,m) and X U is finite, where T(x,m)={x n:n≥n}. (2) A topology space X is metrization if and only if X has a week development G 1,G 2,… Such that for each natural numberi and any two sets U 1,U 2∈G i+1 with non empty intersection then exists a set U∈G i such that U 1∩U 2U.

Let M,V,Q be Lipschitz manifolds, M be a locally flat and compact submanifold of V, V be an open manifold and dim V =dim Q. And let U be an open neighborhood of M in V and Δ n be the n dimensional standard simplex in R n. if f: Δ n×U→Δ n×Q is a LIP immersion and P 1f=P 1, we call f an n dimensional simplex. Let (IM V(M,Q)) n be the set of all f and IM V(M,Q) ={(IM V(M,Q)) n} n≥0. In this paper, we proved that Im v(M,Q) ...

Let M,V,Q be Lipschitz manifolds, M be a locally flat and compact submanifold of V, V be an open manifold and dim V =dim Q. And let U be an open neighborhood of M in V and Δ n be the n dimensional standard simplex in R n. if f: Δ n×U→Δ n×Q is a LIP immersion and P 1f=P 1, we call f an n dimensional simplex. Let (IM V(M,Q)) n be the set of all f and IM V(M,Q) ={(IM V(M,Q)) n} n≥0. In this paper, we proved that Im v(M,Q) together with the face operator i and the degeneracy operater S i defined by us is a semisimplicial complex.