On countable choice and sequential spaces

Under the axiom of choice, every first countable space is a Fréchet-Urysohn space. Although, in its absence even R may fail to be a sequential space.Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of R, are classes of Fréchet-Urysohn or sequential spaces.In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion.Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in R if and only if the axiom of countable choice holds for families of subsets of R, and every metric space has a unique -completion. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim