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On countable choice and sequential spaces

Gonçalo Gutierres

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

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On sequentially closed subsets of the real line in.Kyriakos Keremedis - 2015 - Mathematical Logic Quarterly 61 (1-2):24-31.details We show: iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete. Every infinite subset X of has a countably infinite subset iff every infinite sequentially closed subset of includes an infinite closed subset. The statement “ is sequential” is equivalent to each one of the following propositions: Every sequentially closed subset A of includes a countable cofinal subset C, for every sequentially closed subset (...) Download Export citation Bookmark 1 citation
The Ultrafilter Closure in ZF.Gonçalo Gutierres - 2010 - Mathematical Logic Quarterly 56 (3):331-336.details 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 (...) Download Export citation Bookmark

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