Towards a Natural Proof of Metrization Theorem for Space-Times

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, the proofs of these 1970s results are not natural -- in the sense that they looks like clever tricks, not like a direct consequence of the definitions. Since one of the main objectives of this activity is to come up with useful applications to physics, we definitely desire more natural versions of these proofs. In this paper, we show that fuzzy logic leads to such natural proofs