
details
Author(s)
Gonçalo Gutierres;
Title On countable choice and sequential spaces
Abstract Under the axiom of choice, every first countable space is a FréchetUrysohn space. Although, in its absence
even R may fail to be a sequential space.
Our goal in this paper is to discuss under which settheoretic conditions some topological classes, such as
the first countable spaces, the metric spaces, or the subspaces of R, are classes of FréchetUrysohn 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 σcompletion.
Journal Mathematical Logic Quarterly
Volume 54
Year 2008
Issue 2
Page(s) 145152

