Exponentiable Objects and Function spaces in Lowen Fuzzy Topological Spaces
Abstract
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 τX with C(X,Sier), we explicitly determine the largest splitting topology on this mapping set. Its open weights Φ:τX→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∈τX satisfies μ=bigveeλtriangleleftΦ (constΦ(μ)∧λ), λtriangleleftΦ Longleftrightarrow constΦ(ν)∧λ≤ν (ν∈τX). When this condition holds, YX has underlying set C(X,Y), with topology generated by [Φ,v](f)=Φ(v∘ f). We also obtain a dual closed-set formulation and three applications. Exponentiability implies that τX 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.
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BibTeX
@misc{li2026exponentiable,
title = {{Exponentiable Objects and Function spaces in Lowen Fuzzy Topological Spaces}},
author = {Yongming Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23281},
eprint = {2609.23281},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23281},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exponentiable Objects and Function spaces in Lowen Fuzzy Topological Spaces
AU - Yongming Li
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23281
UR - https://arxiv.org/abs/2609.23281
AB - 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→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∈τ_X satisfies μ=bigvee_λtriangleleftΦ (constΦ(μ)∧λ), λtriangleleftΦ Longleftrightarrow constΦ(ν)∧λ≤ν (ν∈τ_X). When this condition holds, Y^X has underlying set C(X,Y), with topology generated by [Φ,v](f)=Φ(v∘ 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.
ER -