The search for necessary and sufficient conditions for the metrizability of topological spaces in one of the oldest and most productive problems of point set topology. Alexandroff and Urysohn  provided one solution as early as 1923 by imposing special conditions on a sequence of open conversing. Nearly ten years later R.L. Moore chose to begin his classic text on the Foundations of Point Set Theory  with an axiom structure which was a slight variation of the Alexandroff and Urysohn metrizability conditions. After Jones , we now call any space which satisfies Axion 0 and parts 1, 2, 3 of Axiom 1 of  a Moore space. Each metric space is a Moore space, but not conversely, so the search for a metrization theorem became that of determining precisely which Moore spaces are metrizable. The most famous conjecture was that each normal Moore space is metrizable.
KeywordsContinuum Hypothesis Regular Space Paracompact Space Uniform Base Metrization Theory
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