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Sequential topological conditions in ℝ in the absence of the axiom of choice

Gonçalo Gutierres

Mathematical Logic Quarterly 49 (3):293-298 (2003)

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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
On countable choice and sequential spaces.Gonçalo Gutierres - 2008 - Mathematical Logic Quarterly 54 (2):145-152.details 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 (...) Download Export citation Bookmark 2 citations
Properties of the real line and weak forms of the Axiom of Choice.Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Eleftherios Tachtsis - 2005 - Mathematical Logic Quarterly 51 (6):598-609.details We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals. Download Export citation Bookmark 1 citation
(1 other version)On Sequentially Compact Subspaces of without the Axiom of Choice.Kyriakos Keremedis & Eleftherios Tachtsis - 2003 - Notre Dame Journal of Formal Logic 44 (3):175-184.details We show that the property of sequential compactness for subspaces of. Download Export citation Bookmark 2 citations
Sequential topologies and Dedekind finite sets.Jindřich Zapletal - 2022 - Mathematical Logic Quarterly 68 (1):107-109.details It is consistent with ZF $\mathsf {ZF}$ set theory that the Euclidean topology on R $\mathbb {R}$ is not sequential, yet every infinite set of reals contains a countably infinite subset. This answers a question of Gutierres. Download Export citation Bookmark

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