Abstract The ‘the’ in the title hides a subtlety. A metric space induces not one but four topologies - by means of open sets, closed sets, closure, and interior - they just so happen to coincide. The agreement between these four structures arising from a metric function d : X × X → [ 0 , ∞ ] is due to a combination of the metric axioms and the lattice structure of [ 0 , ∞ ] . Further motivation materializes from Lawvere's observation from 1973 to the effect that a (slightly generalized) metric space is a category enriched in [ 0 , ∞ ] . Metric spaces taking values in structures other than [ 0 , ∞ ] are relevant for generalizations of metric spaces and find a natural home in Lawvere's categorical setting. In particular, in recent years quantales emerged as structures occupying an important niche in between [ 0 , ∞ ] and arbitrary monoidal categories. Since a category enriched in a quantale Q is the same thing as a metric space taking values in Q one may ask whether such a thing belongs to algebra or geometry. Further, does the quadruplet of topologies associated to a Q-valued space/category still consist of identical siblings? We propose a litmus test for the geometricity of Q-valued spaces as we investigate these issues.

中文翻译：

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量子值度量空间的拓扑

摘要 标题中的“the”隐藏着微妙之处。度量空间引发的不是一种而是四种拓扑——通过开集、闭集、闭包和内部——它们恰好重合。由度量函数 d 产生的这四种结构之间的一致性：X × X → [ 0 , ∞ ] 是由于度量公理和 [ 0 , ∞ ] 的格结构的组合。进一步的动机体现在 Lawvere 1973 年的观察中，即（稍微广义的）度量空间是一个富含 [0, ∞] 的范畴。在除 [ 0 , ∞ ] 之外的结构中取值的度量空间与度量空间的推广相关，并在 Lawvere 的分类设置中找到了自然的归宿。特别是，近年来，量子作为结构占据了 [ 0 , ∞ ] 和任意幺半群范畴。由于在量子 Q 中丰富的类别与在 Q 中取值的度量空间相同，因此人们可能会问这样的事情是否属于代数或几何。此外，与 Q 值空间/类别相关联的四元组拓扑是否仍然由相同的兄弟姐妹组成？在研究这些问题时，我们建议对 Q 值空间的几何性进行试金石检验。