Moore Spaces and Uniform Spaces
F. Burton Jones · Proceedings of the American Mathematical Society · 1958
Although arising from different motivationsMoore spaces2 and uniform spaces [7] are generalizations of metric spaces which have considerable similarity.This can be seen from the following characterizations by means of open coverings:Let .S be a regular Hausdorff space.If there exists a family {G} of open coverings of S such that (a) {G} is countable and3 (b) if U is an open set and pEU, there is an element G of {G} such that GP*E U, then 5 is a Moore space and conversely [6].If there exists a family {G} of open coverings of 5 such that (a') if Gi and G2 are elements of {G} there is an element C73 of {G} which is a star refinement of both Gi and G2 and (b) if U is an open set and pEU, there is an element G of {c7} such that GP*E U, then 5 is a uniform space4 and conversely [5].Of course if a {G} exists so that all three conditions (a), (a') and (b) hold true simultaneously, 5 is metric.6Except for the obvious fact that the Hausdorff first countability axiom must hold true for Moore spaces while it need not do so for uniform spaces, it is by no means obvious that they differ otherwise.L. F. McAuley has given an example of a normal semi-metric Hausdorff space 5 which is not a Moore space [3].Being normal, 5 is both regular and uniform and being semi-metric, the first countability axiom holds true for S. I give below an example of a locally connected, connected, complete Moore space6 containing a point p at which it is not completely regular.Hence it is not uniform [2 or 5].This space is obtained by piecing together along their boundaries adjacent terms of a simple sequence