| Résume | Metrizable spaces are among the oldest and most prominent concepts in numerous areas of mathematics. Consequently, many of their properties are well-known.
Some of these properties are quite old. For instance, Cantor's intersection theorem from the early 20th century provides insights into complete metrics and offers a characterization of completeness. The Nagata-Smirnov metrization theorem, which delivers the first complete characterization of metrizability in terms of other topological properties, dates back to the 1950s.
However, even slight modifications to these results lead to conjectures that are both natural and easy to formulate — so straightforward, in fact, that they are understandable to anyone with a basic knowledge in mathematics. Yet, it remains unknown whether these conjectures hold.
In this talk, I aim to survey what remains to be studied about metrics and their generalizations. I will present some of the most natural and fundamental open questions in this area, discuss their significance, and share what is currently known about them.
The material for this talk is largely based on joint work with Luca Motto Ros. |
