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
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
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 σ-completion.
Mathematical Logic Quarterly