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Author(s)
Gonçalo Gutierres;
Title The Ultrafilter Closure in ZF
Abstract It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF (ZermeloFraenkel set theory without
the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultrafilter Theorem is equivalent to the fact that u_{X} = k_{X} for every topological space X, where k is the usual Kuratowski Closure operator and u is the Ultrafilter Closure with
u_{X}(A) := {x ∈ X : (∃U ultrafilter in X)[U converges to x and A ∈ U]}.
However, it is possible to build a topological space X for which u_{X} \neq k_{X}, but the open sets are characterized by the ultrafilter convergence. To do so, it is proved that if every set has a free ultrafilter then the Axiom of Countable Choice holds for families of nonempty 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 T_{0}spaces or {R}.
Preprint series Prépublicações do Departamento de Matemática da Universidade de Coimbra
Issue 0837
Year 2008

