中文摘要：

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.

英文摘要：

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.