What if all topological spaces were metrisable?

08/03/2019, 14:00 - 15:00 in MAP/0G/018

Weiss, Ittay (Portsmouth)


Abstract:

In 1906 Fréchet introduced the concept of metric space in his PhD dissertation in what was at the time a decisive step toward abstractness. The definition itself, simple and intuitive, released analytic and geometric ideas from the confines of Euclidean spaces and allowed a systematic study of function spaces, eventually leading to many modern developments. However, it was not long before the applicability boundary of the formalism was reached as topological issues were slowly being crystallised; new spaces were popping up which failed to possess a metric but still had a notion of convergence. Metric space theory and topology became two quite distinctly flavoured areas, with metrisable spaces viewed as a very special class of topological spaces. In 1997 Flagg introduced the concept of a value quantale, a certain ordered structure distilling key properties of the structure of [0,∞]. Allowing a metric space to take values in a value quantale rather than insisting on the distances landing in [0,∞] bears a remarkable consequence: All topological spaces are metrisable. Not even assuming knowledge of what a metric space is, I will sketch the proof of the metrisability of all topological spaces, highlight the resulting (counter-historical) unification of topology and geometry under a single formalism, and proceed to consider some of its consequences from recent research.