The topology of a quantale valued metric space

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