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On sequentially closed subsets of the real line in ZF
Author(s) -
Keremedis Kyriakos
Publication year - 2015
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201300008
Subject(s) - mathematics , countable set , closed set , separable space , uncountable set , metric space , cantor set , combinatorics , subspace topology , discrete mathematics , nowhere dense set , statement (logic) , real line , complete metric space , set (abstract data type) , mathematical analysis , computer science , political science , law , programming language
We show: (i) CAC 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. (ii) Every infinite subset X of R has a countably infinite subset iff every infinite sequentially closed subset of R includes an infinite closed subset. (iii) The statement “ R is sequential” is equivalent to each one of the following propositions:(a) Every sequentially closed subset A of R includes a countable cofinal subset C , (b) for every sequentially closed subset A of R ,A ¯ ∖ Ais a meager subset of A ¯ , (c) for every sequentially closed subset A of R ,A ¯ ∖ A ¯ ≠ A ¯ , (d) every sequentially closed subset of R is separable, (e) every sequentially closed subset of R is Cantor complete, (f) every complete subspace of R is Cantor complete.
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