Sober topological spaces valued in a quantale

The notion of closedness and D-connectedness in Quantale-valued approach spaces

In this paper, we characterize the local T0 and T1 separation axioms for quantale-valued gauge space, show how these concepts are related to each other and apply them to L-approach space and L-approach system. Furthermore, we give the characterization of a closed point and D-connectedness in quantale-valued gauge space. Finally, we compare all these concepts with other.

Sobriety of quantale-valued cotopological spaces

For each commutative and integral quantale, making use of the fuzzy order between closed sets, a theory of sobriety for quantale-valued cotopological spaces is established based on irreducible closed sets.

Quantale-valued Approach Spaces via Closure and Convergence

For a quantale $\mathsf{V}$ we introduce $\mathsf{V}$-approach spaces via $\mathsf{V}$-valued point-set-distance functions and, when $\mathsf{V}$ is completely distributive, characterize them in terms of both, so-called closure towers and ultrafilter convergence relations. When $\mathsf{V}$ is the two-element chain $\mathsf{2}$, the extended real half-line $[0,\infty]$, or the quantale ${\bfΔ}$ of distance distribution functions, the general setting produces known and new results on topological spaces, approach spaces, and the only recently considered probabilistic approach spaces, as well as …

$L$-凸空间的 sober 性刻画

The characterization for the sobriety of $L$-convex spaces

以交换整量子化格 $L$ 为真值表，研究分层 $L$-凸空间 sober 性的刻画，证明 sober 化等价于拟同胚存在，且 sober 性等价于在分层 $S_0$ $L$-凸空间范畴中为严格内射对象。

拟一致结构、幂集单子与定性鲁棒性分析的度量化

Metrization of Quasi-Uniformities, Powerset Monads, and Qualitative Robustness Analysis

该研究建立了拟一致空间范畴与 quantale 值度量空间范畴的度量化等价定理，细化了拓扑空间的 quantale 度量化构造，将定量鲁棒性分析所用的 Hausdorff-Smyth 单子提升至拟一致空间，构建了三者的统一范畴框架。

量子值背景下通过闭包空间的域表示

Representations of domains via closure spaces in the quantale-valued setting

本研究以交换单位量子$L$为真值表，通过引入插值广义$L$-闭包空间和稠密子空间，建立了$L$-闭包空间与连续/代数$L$-dcpo之间的范畴等价，实现了$L$-域的表示。