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A topological space S is said to be a semi-metric space provided that there is a distance function (or "semi-metric") d defined for S such that, if each of x and y is a point of S, then (1) d(x, y)=d(y, x), (2) d(x, y) 2:0 and d(x, y) =0 only in case x = y, and (3) the topology of S is invariant with respect to the distance function d—i.e., if p is a limit point of a subset M of S, then p is a distance limit point of M and conversely. If x^.S and c>0, {y:yGS and d(x, y) <c} is a c-neighborhood (or "c-sphere") of the point x. For the purpose of constructing semi-metric spaces the following characterization from [l] is useful. A necessary and sufficient condition that the topological space S be semi-metric is that there exist a collection of open sets, jgn(x): x G S, n = 1, 2, 3, • • • }, such that (1) for each x (E. S, {gn(x)} "„i is a nonincreasing sequence which forms a local base for the topology at x, and (2) if y(ES and x is a point sequence in S such that, for each m, yGgm(xm), then x converges to y. A good source of information on some of the work that has been done on semi-metric spaces is to be found in [4]. As is pointed out there by F. B. Jones, a regular semi-metric space is not too far removed from a Moore space (every Moore space being a regular semimetric space) in that most of the theorems of Moore's book hold in a regular semi-metric space (with the addition of an appropriate completeness axiom, local connectedness, etc., as needed)—in fact a strongly complete regular semi-metric space is a Moore space [l]. In [4] Morton Brown noted several open questions including: "Does every semi-metric space have a semi-metric under which all spheres are open?" It is well known that every Moore space does have such a semi-metric; however, the general question is answered in the negative by the following theorem.

A regular semi-metric space for which there is no semi-metric under which all spheres are open