Metrization Theorem for Space-Times: From Urysohn\u27s Problem Towards Physically Useful Constructive Mathematics

In the early 1920s, Pavel Urysohn proved his famous lemma (sometimes referred to as first non-trivial result of point set topology ). Among other applications, this lemma was instrumental in proving that under reasonable conditions, every topological space can be metrized. A few years before that, in 1919, a complex mathematical theory was experimentally proven to be extremely useful in the description of real world phenomena: namely, during a solar eclipse, General Relativity theory -- that uses pseudo-Riemann spaces to describe space-time -- has been (spectacularly) experimentally confirmed. Motivated by this success, Urysohn started working on an extension of his lemma and of the metrization theorem to (causality-)ordered topological spaces and corresponding pseudo-metrics. After Urysohn\u27s early death in 1924, this activity was continued in Russia by his student Vadim Efremovich, Efremovich\u27s student Revolt Pimenov, and by Pimenov\u27s students (and also by H. Busemann in the US and by E. Kronheimer and R. Penrose in the UK). By the 1970s, reasonably general space-time versions of Urysohn\u27s lemma and metrization theorem have been proven. However, these 1970s results are not constructive. Since one of the main objectives of this activity is to come up with useful applications to physics, we definitely need constructive versions of these theorems -- versions in which we not only claim the theoretical existence of a pseudo-metric, but we also provide an algorithm enabling the physicist to generate such a metric based on empirical data about the causality relation. An additional difficulty here is that for this algorithm to be useful, we need a physically relevant constructive description of a causality-type ordering relation. In this paper, we propose such a description and show that for this description, a combination of the existing constructive ideas with the known (non-constructive) proof leads to successful constructive space-time versions of the Urysohn\u27s lemma and of the metrization theorem