Co-RS-compact Topologies

A topoogy R(r)is constructed from a given topology r on a set X.R(r)iscoarser than r,and the following are some results based on this topology:1.Continuity and RS-continuity are equivalent if the codomain is retopologized by R(τ).2.Each topological spase has a coarser extremely disconnected topology3.The class of semi-open sets with respect to R(r)is a topology4.T_2 and semi-T_2 properties are equivalent on a space whose topology is R(r).5.Minboal R_0-spaces are RS-compact.6.MaxAnalα-compact spaces are RS-compact.     

Fuzzy度量化——嵌入理论的一个应用

A fuzzy Pseudo-metric space is called fuzzy metric space iff it is sub-T₀ space. Suppose that (X,F) is a fuzzy topological space. Consider a relation ～between ordinary points of X: x～y iff for each λ≠0,x_λ∈y_λ and y_λ∈x_λ. The relation～is an equivalent relation. The corresponding fuzzy quotien space is a sub-T0 space and is called the associated sub-T₀ space of (X,F). in this paper, we establish the following theorem via fzzzy imbedding theory:Theorem Suppose that (X,F) is a fuzzy topological space with countable topological base. Then (1)(X,F) is fuzzy metrizable iff it is a fuzzy completely regular and sub-T₀ space.(2)(X,F) is a fuzzy completely regular space,then its associated sub-T₀ space is fuzzy metrizable.     

ON CONDITIONS FOR THE NON LOCALLY CONVEX LINEAR TOPOLOGICAL SPACE TO HAVE THE H.-B. EXTENSION PROPERTY

In this paper, the conditions for the non lcally convex topological vector space to have the H,-B. extension property is discussed, and the following three results are proved; (1)A closed subspace E0 of a linear topological space E to have the H.-B. property if and only if for every closed hyperplane of E0 is weakly closed, (2) A locally bounded linear topological space (E,τo)to have the H.-B extension property if and only if for every closed subspace E0 of E, the weak topology σ(E0,E^*0)属于τ1|E0, where τ1 is the finest locally convex topology on E which is coarser then τ0. (3)Let E be separated and let E be the completion of E. If every closed subspace E0 of E is the complete hull of E0∩E,then E has H.-B. extension property if and only if E has H.-B. extension property.     

$C(X)$上的开点与紧开拓扑

In this note we define a new topology on C(X),the set of all real-valued continuous functions on a Tychonoff space X.The new topology on C(X) is the topology having subbase open sets of both kinds:[f,C,ε[={g E C(X):|f(x)-g(x)| ε for every x∈C} and[U,r]~-={g∈C(X):g~(-1)(r)∩U≠φ},where f∈C(X),C∈KC(X)={nonempty compact subsets of X},ε 0,while U is an open subset of X and r∈R.The space C(X) equipped with the new topology T_(kh) which is stated above is denoted by C_(kh)(X).Denote X_0={x∈X:x is an isolated point of X} and X_c={x∈X:x has a compact neighborhood in X}.We show that if X is a Tychonoff space such that X_0=X_c,then the following statements are equivalent:(1) X_0 is G_δ-dense in X;(2) C_(kh)(X) is regular;(3) C_(kh)(X) is Tychonoff;(4) C_(kh)(X) is a topological group.We also show that if X is a Tychonoff space such that X_0=X_c and C_(kh)(X) is regular space with countable pseudocharacter,then X is σ-compact.If X is a metrizable hemicompact countable space,then C_(kh)(X) is first countable.     

Hilbert-Schmidt类上的k-拟亚正规算子

An operator T acting on a Hilbert space H is called k- quasihyponormal if {T} =T^*k (T^*T-TT^*)T^k≥0 . Let A and B be two operators on H, one can obtain an operator τ =τ_AB on the class l₂ of all Hilbert-Schmidt operators on H in such a way that τ(X)=AXB for every X∈l₂. In this note the authors show that     

Equivariant Mapping Class Group and Orbit Braid Group

Motivated by the work of Birman about the relationship between mapping class groups and braid groups, the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper. Their construction is based on the exact sequence given by the fibration F₀^GM → F(M/G, n). The conclusion is closely connected with the braid group of the quotient space. Comparing with the situation without the group action, there is a big difference when the quotient space is T².     

Heisenberg群上一类左不变LPDO的局部基本解及局部可解性

In this paper it is proved that local fundamental solution exists in some space W^m(H_n) (m∈Z), if the left invariant differential operator on the Heisenberg group H_n satisfies certain condition. The main results are:l.Let L be a left invariant differential operator on H_n. If there exist R≥0, r,s∈R and operators {B_λ|λ∈Γ_R} ∈V^s(Γ_R, M^r) such that, for almost all λ∈Γ_R, B_λ is the right inverse of Ⅱ_λ(L), then there exists E∈W^m(H_n) (when m≥0 or m even) or E∈W^m-1(H_n) (when m<0 and odd) such that LE =δ(near the origie) Where m=min([r],-[2s]-n-2); 2. Let L(W,T) be of the form (3.1). If there exist R≥0 and r,s∈R such that when |λ|≥R,(?) and C_λ≥ C|λ|^x(C>0), then the same conclusion as above holds with m=min(-[2r]-n-2,[-2s]-n-2).     

质环的求导和交换性

In this paper, we generalize some corresponding results of [1-4]. We obtain the main results as the following:Theorem 1 Let R be a prime ring of characteristic not 2 with nontrivial derivations d₁,d₂ and let U be a nonzero ideal of R . If C is the center of R, then the following conditions are equivalent : (i)d₁,d₂(x)∈C for all x∈U; (ii) [d₁(x),d₂(y)]∈C for all x,y∈U; (iii) d₁(x)d₂(y)+d₂(x)d₁(y)∈C for all x,y∈U; (iv) R is commutative.Theorem 2 Let R be a prime ring with nontrivial derivations d₁,d₂,…, d_n and U be a nonzero ideal of R. Let C be the center of R. If d₁(x₁)d₂(x₂)…d_n(x_n)∈C for all x₁, x₂…x_n)∈U, then R is commutative.     

Cohn环与Grothendieck群

We introduce three classes of cohn rings C₁,C₂ and C₃). Let R and S are rings,φ : R→S is a ring homomorphism. We prove that R∈C_i if and only if R/J(R) ∈ C_i. In We also discuss the relation between the     

关于θ-加细性

In this paper the preimage problem and hereditarily of θ-refinability are discussed.The main results are as followsi (1) The preimage of weakly δθ-refinable (δθ-refinable, weakly θ-refinable) space under a continuous closed mapping with f^-1(y) Lindelof is weakly δθ-refinable (δθ-refinable, weakly θ-refinable) space; (2) The θ-refinable space with closed G_δ is a hereditarily θ-refinable space.