Towards a more natural proof of metrization theorem for space-times

In the 1920s, Pavel Urysohn proved his famous lemma (sometimes referred to as “first non-trivial result of point set topology”). This lemma was instrumental in proving that under reasonable conditions, every topological space can be metrized. Motivated by the success of pseudo-metric spaces in General Relativity, Urysohn started working on an extension of his lemma and of the metrization theorem to (causality-)ordered topological spaces and corresponding pseudo-metrics. By the 1970s, general space-time versions of Urysohn's lemma and metrization theorem have been proven. However, the proofs of these results are not natural - 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 desire more natural versions of these proofs. In this paper, we show that fuzzy logic leads to such natural proofs.