Axioms That Define Semi-metric, Moore, and Metric Spaces

In [l] L. F. McAuley asked the following question: is it possible to partition . . . Moore's metrization theorem into three or more parts which begins with a condition for a topological space and which ends with a condition for a metrizable space, but with necessary and sufficient conditions somewhere between these extremes for semi-metric and Moore spaces? Axiom Z, stated below, is such a partitioning. The notation "Axiom Z¿" denotes parts (1), (2), • • • , (i) of Axiom Z. In §1 it is proved in Theorems 1, 2, and 3, respectively, that a necessary and sufficient condition for a topological space to be semi-metrizable, a Moore space, and metrizable is that it satisfy Axiom Z2, Axiom Z3, and Axiom Z4 respectively. A counter-example is given in §2 which shows that the argument for the statement a Moore space is a semimetric topological space in Theorem 6.2 in [l] is not correct. Finally, in §3 it is shown that part (3) of Theorem 2 in [2 ] can be changed so that the resulting statement is equivalent to a Moore space. Definitions are given in [l]. Definition. If {/„} denotes a sequence such that for each natural number n, /„ denotes a collection of neighborhoods covering a point set M, then the sequence {-B¿}, where i denotes a natural number, is said to be a basic refinement of {Jn} for M provided that with each point pin M there is associated a sequence {bt(p)} such that for each i: (1) bi(p) is a neighborhood in {/„}, (2) bi+i(p) is a subset of b¡(p), (3) p is the only point common to {bi(p)}, and (4) Bt denotes the collection of all neighborhoods bt(p) for all points in M.