TOPOLOGICAL PRESERVATION BY QUASI-PSEUDOMETRIC AGGREGATION FUNCTIONS

Aggregation functions play a fundamental role in mathematics, particularly in the combination of multiple measurements into a single quantity. They also allow the construction of a single mathematical structure of the same type from a family of given structures. Early work in this area dates back to Boršík and Doboš, who studied the aggregation of metrics. Their results motivated further research on the aggregation of more general structures, including quasi-pseudometrics, fuzzy quasi-pseudometrics, and indistinguishability operators.

All these structures induce a topology on their underlying sets, which raises a natural question: does the aggregation process preserve the associated topological properties? In particular, one may ask whether the topology generated by an aggregated (quasi-)(pseudo)metric coincides with the product topology of the individual spaces. For metric aggregation functions on products, this problem was solved by Boršík and Doboš through the introduction and characterization of strongly metric aggregation functions.

In this work, we extend their approach to the broader setting of quasi-pseudometric aggregation functions on products. We introduce the notion of strongly (quasi-)(pseudo)metric aggregation functions on products and provide a complete characterization of this class. A function is said to be strongly aggregating if, for every family of (quasi-)(pseudo)metric spaces, the topology induced by the aggregated (quasi-)(pseudo)metric agrees with the corresponding product topology.

As a key illustration of our results, we prove that a (quasi-)metric aggregation function on products is strongly aggregating if and only if it is continuous at the zero vector. This condition yields a concise and transparent topological characterization.

In conclusion, our results generalize the classical theory of strongly metric aggregation functions to the quasi-pseudometric framework and establish explicit relationships between the different classes of strongly quasi-pseudometric aggregation functions on products.