关于Lowen空间指数对象的一点注记

A NOTE ON EXPONENTIAL LOWEN SPACES

Ｌ－拓扑空间（Ｘ，△）称为一 Ｌｏｗｅｎ空间若△有一组由层特征函数构成的基，即△中形如ａ∧Ｕ，ａ∈Ｌ，Ｕ Ｘ的元素构成△的一组基．若Ｌ＝［０，１］；则（Ｘ，△）是一Ｌｏｗｅｎ空间当且仅当（Ｘ，△）是一 Ｌｏｗｅｎ意义下的ｆｕｚｚｙ邻域空间．通过在函数空间上引入适当的Ｌ－拓扑结构，证明了若０∈Ｌ是一素元并且Ｌｏｗｅｎ空间（Ｘ，△）的开集格是一连续格，则（Ｘ，△）是Ｌｏｗｅｎ空间范畴中一指数对象．特别地，若一ｆｕｚｚｙ邻域空间的开集格连续，则它是ＦＮＳ中一指数对象．

An L-topological space (X,△) is called a Lowen space if △ has a basis consisting of leveled characteristic functions, that is to say, the elements in △ of the form a ∧ U, a∈L,U X, is a basis for △. In the case L = [0,1], (X, △) is a Lowen space if and only if (X, △) is a fuzzy neighborhood space in the sense of Lowen. By introducing some natural L-topologies for function spaces it is proved in this paper that if 0 ∈ L is a prime then a Lowen space (X, △) is exponential in the category of Lowen spaces provided that its open-set lattice is continuous. Particularly a fuzzy neighbourhood space is exponential in FNS if its open-set lattice is continuous.