发表机构
School of Mathematics and Statistics, Shaanxi Normal University（陕西师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在分层Lowen模糊拓扑空间范畴中刻画可指数对象，给出最大分裂拓扑及内在条件，并证明可指数性蕴含连续格但反之不成立，且经典空间可指数当且仅当其诱导模糊空间可指数，同时Lowen紧、强模糊紧和N-紧Hausdorff空间均可指数。
AI 中文摘要
我们研究了在\\(\I=[0,1]\\)上的分层Lowen模糊拓扑空间范畴中的可指数对象与函数空间。利用Lowen模糊Sierpiński对象\\(\Sier\\)（它将\\(\tau_X\\)与\\(C(X,\Sier)\\)等同起来），我们明确确定了该映射集合上最大的分裂拓扑。其开权\\(\Phi:\tau_X\to\I\\)恰好是满足Scott连续性和由\\(\Sier\\)的有限幂诱导的有限层相容条件的那些权。这给出了一个内在刻画：\\(X\\)是可指数的当且仅当每个\\(\mu\in\tau_X\\)满足\\[ \mu=\bigvee_{\lambda\triangleleft\Phi} (\const{\Phi(\mu)}\wedge\lambda), \qquad \lambda\triangleleft\Phi \Longleftrightarrow \const{\Phi(\nu)}\wedge\lambda\leq\nu \quad(\nu\in\tau_X). \\] 当此条件成立时，\\(Y^X\\)的底层集合为\\(C(X,Y)\\)，其拓扑由\\([\Phi,v](f)=\Phi(v\circ f)\\)生成。我们还得到了一个对偶的闭集表述和三个应用。可指数性蕴含\\(\tau_X\\)是连续格，尽管其逆命题不成立。此外，一个经典空间\\(X\\)是可指数的当且仅当其诱导模糊空间\\(\omega X\\)在整个分层Lowen范畴中是可指数的。最后，Lowen紧、强模糊紧和\\(N\\)-紧Hausdorff空间都是可指数的。
英文摘要
We study exponentiable objects and function spaces in the category of stratified Lowen fuzzy topological spaces over \(\I=[0,1]\). Using the Lowen fuzzy Sierpiński object \(\Sier\), which identifies \(τ_X\) with \(C(X,\Sier)\), we explicitly determine the largest splitting topology on this mapping set. Its open weights \(Φ:τ_X\to\I\) are precisely those satisfying Scott continuity and a finite-tier compatibility condition induced by finite powers of \(\Sier\). This yields an intrinsic characterization: \(X\) is exponentiable if and only if every \(μ\inτ_X\) satisfies \[ μ=\bigvee_{λ\triangleleftΦ} (\const{Φ(μ)}\wedgeλ), \qquad λ\triangleleftΦ \Longleftrightarrow \const{Φ(ν)}\wedgeλ\leqν \quad(ν\inτ_X). \] When this condition holds, \(Y^X\) has underlying set \(C(X,Y)\), with topology generated by \([Φ,v](f)=Φ(v\circ f)\). We also obtain a dual closed-set formulation and three applications. Exponentiability implies that \(τ_X\) is a continuous lattice, although the converse fails. Moreover, a classical space \(X\) is exponentiable exactly when its induced fuzzy space \(ωX\) is exponentiable in the entire stratified Lowen category. Finally, Lowen compact, strongly fuzzy compact, and \(N\)-compact Hausdorff spaces are exponentiable.
