Let E and F be two vector spaces in duality with respect to the bilinear pairing 〈,〉. The weak (Mackey,strong) topology on E will be denoted by σ(E,F) (τ(E,F),β(E,F)). In this paper,we show that AE is σ(E,F) K b...Let E and F be two vector spaces in duality with respect to the bilinear pairing 〈,〉. The weak (Mackey,strong) topology on E will be denoted by σ(E,F) (τ(E,F),β(E,F)). In this paper,we show that AE is σ(E,F) K bounded subset if and only if A is β(E,F) K bounded subset. If τ(E′,E) is a quasi barrelled space,we also give a few characterizations for (E,T) to be A space.展开更多
Zhao and Ho asked in a recent paper that for each T_0 space X, whether KB(X)(the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and La...Zhao and Ho asked in a recent paper that for each T_0 space X, whether KB(X)(the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Top_κ of the category Top_0 of T_0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Top_κ.展开更多
文摘Let E and F be two vector spaces in duality with respect to the bilinear pairing 〈,〉. The weak (Mackey,strong) topology on E will be denoted by σ(E,F) (τ(E,F),β(E,F)). In this paper,we show that AE is σ(E,F) K bounded subset if and only if A is β(E,F) K bounded subset. If τ(E′,E) is a quasi barrelled space,we also give a few characterizations for (E,T) to be A space.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11531009 and 11871320)the Fundamental Research Funds for the Central Universities(Grant No.GK201803002)
文摘Zhao and Ho asked in a recent paper that for each T_0 space X, whether KB(X)(the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Top_κ of the category Top_0 of T_0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Top_κ.
