Abstract:Let $X$ be a uniform space of uniform weight $\mu $. It is shown that if every open covering, of power at most $\mu $, is uniform, then $X$ is fine. Furthermore, an $\omega _\mu $-metric space is fine, provided that every finite open covering is uniform.
Keywords: uniform space, uniform weight, fine uniformity, uniformly locally finite, $\omega _\mu $-additive space, $\omega _\mu $-metric space
AMS Subject Classification: Primary 54E15; Secondary 54A25, 54A35