Topological preservation under aggregation of quantale-valued metric spaces
Aggregation functions provide a general mechanism for combining multiple inputs into a single output and constitute a fundamental tool in information fusion. Beyond the classical numerical setting, aggregation can also be considered at the level of mathematical structures, where the main question is to determine which properties of the original structures are preserved by the aggregation process. In this paper, we study the preservation of topology within a unified framework based on continuous quantales and quantale-enriched categories. We introduce strongly preserving functions, for which the natural topology induced by a $\mathscr V$-metric space coincides with that induced after aggregation. We decompose this property into its upper and lower versions and show that upper strong preservation is equivalent to Scott continuity at the unit of the domain quantale, whereas lower strong preservation is characterized in terms of the fibre of the unit and the Scott upper semicontinuity of the inverse multifunction. Combining these results, we obtain a complete characterization of strongly preserving functions, together with analogous results for separated, symmetric, and separated and symmetric $\mathscr V$-metric spaces. Finally, we specialize the general theory to functions from $\mathsf P_+^{\,I}$ to $\mathsf P_+$, obtaining characterizations of strongly extended quasi-pseudometric and pseudometric aggregation functions on Cartesian products and on common underlying sets, together with corresponding results for extended quasi-metric and metric aggregation functions on Cartesian products.