Remarks on Metrizability

In connection with those paragraphs of my paper Applications of the theory of Boolean rings to general topology (Trans. Amer. Math. Soc. vol. 41 (1937) pp. 375-481) dealing with regular spaces, I have long been curious to know whether certain results proved there could be used to obtain the well known theorem that a separable (Hausdorff) space is metrizable if (and only if) it is regular. Since a positive answer to the question thus posed may have some interest from a methodological point of view, I communicate a demonstration here. The essential step in this demonstration even has some intrinsic interest, consisting as it does in the proof of new facts about dissectionspaces and the related maps. However, as a proof of the metrizability theorem this discussion is not as simple or as direct as the now classical proof of Tychonofï and Urysohn—which, it may be recalled, consists in showing, first, tha t a separable regular space is normal and, second, that a separable normal space is metrizable. As a direct corollary of theorems established in our paper cited above, we may state the following result.