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. (shrink)

It is well known that, in a topological space, the open sets can be characterized using ?lter convergence. In ZF , we cannot replace filters by ultrafilters. It is proven that the ultra?lter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultra?lter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski closure operator (...) and u is the Ultra?lter Closure with uX := {x ∈ X: [U converges to x and A ∈ U ]}. However, it is possible to built a topological space X for which uX ≠ kX, but the open sets are characterized by the ultra?lter convergence. To do so, it is proved that if every set has a free ultra?lter, then the Axiom of Countable Choice holds for families of non-empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0-spaces or {ℝ}. (shrink)