格值诱导空间中的权、特征及稠密度

WEIGHT,CHARACTER AND DENSITY IN LATTICE-VALUED INDUCED SPACES

利用格值诱导空间中内部算子的层次刻划,得出了格值诱导空间与其底空间之间权、特征及稠密度的三组不等式,从而对一类较广泛的Fuzzy格(即ω-生成的Fuzzy格),证明了诱导空间是第二可数(第一可数、可分)空间,当且仅当底空间是第二可数(第一可数、可分)空间.并举例说明,对ω_1-生成的Fuzzy格,上述性质不必成立.同时,给出了Fuzzy Smirnovhagata度量化定理必要条件不必成立的一个简单例子.

In this paper, three inequalities about weight, character and density between induced space and its underlying space are discribed, by a lever characterization of interior operators in induced spaces. Based on these results, we prove that induced space is second-countable (first-countable, separatable respectively) if and only if its underlying space is second-countable (first-countable , separatable respectively) for an important type of fuzzy lattices (i. e. co-generated fuzzy lattices). Moreover, for ω1-generated lattices, we construct two examples to show that the assertion above result does not hold. Meanwhile, we contruct a simple example in which fuzzy Smirnov-Nagata metrization theorem does not hold.