On countable choice and sequential spaces

Mathematical Logic Quarterly 54 (2):145-152 (2008)
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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 equation image-completion



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On Sequentially Closed Subsets of the Real Line in ZF.Kyriakos Keremedis - 2015 - Mathematical Logic Quarterly 61 (1-2):24-31.
The Ultrafilter Closure in ZF.Gonçalo Gutierres - 2010 - Mathematical Logic Quarterly 56 (3):331-336.

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