一句话：在 V 为空间余框架的较弱条件下，无选择公理地证明 V-拓扑空间范畴是 Set 上的拓扑范畴。
对于 quantale V，V 值拓扑空间可定义为 V 值闭包空间的全子范畴，且其闭包运算保持有限并。Barr 曾将拓扑空间刻画为超滤子单子从映射到集合关系之 lax 扩张的 lax 代数；新近工作表明，当 V 完全分配时，V-拓扑空间也可由超滤子单子到 V 值关系的 lax 扩张刻画，从而 V-Top 是 Set 上的拓扑范畴。本文去掉了完全分配性假设，给出无选择公理的证明：当 V 是空间余框架时，V-Top 是 Set 上的拓扑范畴。特别地，当 V 是连续格时，空间余框架条件可在构造性意义下推出完全分配性，因而在承认选择公理时也给出通常意义的完全分配性。
For a quantale ${\sf{V}}$, the category $\sf V$-${\bf Top}$ of ${\sf{V}}$-valued topological spaces may be introduced as a full subcategory of those ${\sf{V}}$-valued closure spaces whose closure operation preserves finite joins. In generalization of Barr's characterization of topological spaces as the lax algebras of a lax extension of the ultrafilter monad from maps to relations of sets, for ${\sf{V}}$ completely distributive, ${\sf{V}}$-topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to ${\sf{V}}$-valued relations. As a consequence, ${\sf{V}}$-$\bf Top$ is seen to be a topological category over $\bf Set$, provided that ${\sf{V}}$ is completely distributive. In this paper we give a choice-free proof that ${\sf{V}}$-$\bf Top$ is a topological category over $\bf Set$ under the considerably milder provision that ${\sf{V}}$ be a spatial coframe. When ${\sf{V}}$ is a continuous lattice, that provision yields complete distributivity of ${\sf{V}}$ in the constructive sense, hence also in the ordinary sense whenever the Axiom of Choice is granted.
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@article{LaiT16,
title = {A Note on the Topologicity of Quantale-Valued Topological Spaces},
author = {Hongliang Lai and Walter Tholen},
journal = {Logical Methods in Computer Science},
volume = {Volume 13, Issue 3},
year = {2017},
doi = {10.23638/lmcs-13(3:12)2017}
}