In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces（CSS） by semi-continuous functions. We mainly show that：（1）Let X be a space. Then the following state...In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces（CSS） by semi-continuous functions. We mainly show that：（1）Let X be a space. Then the following statements are equivalent：（i） X is almost completely regular.（ii） Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.（iii） If g, h ： X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f ： X → I such that g ≤ f ≤ h;and（2） Let X be a space. Then the following statements are equivalent：（a） X is CSS;（b） There is an operator U assigning to a decreasing sequence of compact sets（Fj）j∈N,a decreasing sequence of open sets（U（n,（Fj）））n∈N such that（b1） Fn■U（n,（Fj）） for each n ∈ N;（b2）∩n∈NU（n,（Fj）） =∩n∈NFn;（b3） Given two decreasing sequences of compact sets（Fj）j∈N and（Ej）j∈N such that Fn■Enfor each n ∈ N, then U（n,（Fj））■U（n,（Ej）） for each n ∈ N;（c） There is an operator Φ ： LCL（X, I） → USC（X, I） such that, for any h ∈ LCL（X, I）,0 Φ（h） h, and 0 〈 Φ（h）（x） 〈 h（x） whenever h（x） 〉 0.展开更多
Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The gen...Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement <sup>*</sup>X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group <span style="white-space:nowrap;">Γ of the locally compact Abelian group G is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.展开更多
基金Supported by the Project of Young Creative Talents of Guangdong Province(Grant No.2014KQNCX161)the Ph D Start-up Fund of Natural Science Foundation of Guangdong Province(Grant No.2014A030310187)+3 种基金the National Natural Science Foundation of China(Grant Nos.11526158613792111471153)the Research Fund for Higher Education of Fujian Province of China(Grant No.2013J01029)
文摘In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces（CSS） by semi-continuous functions. We mainly show that：（1）Let X be a space. Then the following statements are equivalent：（i） X is almost completely regular.（ii） Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.（iii） If g, h ： X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f ： X → I such that g ≤ f ≤ h;and（2） Let X be a space. Then the following statements are equivalent：（a） X is CSS;（b） There is an operator U assigning to a decreasing sequence of compact sets（Fj）j∈N,a decreasing sequence of open sets（U（n,（Fj）））n∈N such that（b1） Fn■U（n,（Fj）） for each n ∈ N;（b2）∩n∈NU（n,（Fj）） =∩n∈NFn;（b3） Given two decreasing sequences of compact sets（Fj）j∈N and（Ej）j∈N such that Fn■Enfor each n ∈ N, then U（n,（Fj））■U（n,（Ej）） for each n ∈ N;（c） There is an operator Φ ： LCL（X, I） → USC（X, I） such that, for any h ∈ LCL（X, I）,0 Φ（h） h, and 0 〈 Φ（h）（x） 〈 h（x） whenever h（x） 〉 0.
文摘Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement <sup>*</sup>X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group <span style="white-space:nowrap;">Γ of the locally compact Abelian group G is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.
