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https://hdl.handle.net/10316/8212
Title: | On countable choice and sequential spaces | Authors: | Gutierres, Gonçalo | Issue Date: | 2008 | Citation: | MLQ. 54:2 (2008) 145-152 | Abstract: | 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 -completion. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) | URI: | https://hdl.handle.net/10316/8212 | DOI: | 10.1002/malq.200710018 | Rights: | openAccess |
Appears in Collections: | FCTUC Matemática - Artigos em Revistas Internacionais |
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