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Sequential topological conditions in ℝ in the absence of the axiom of choice
Author(s) -
Gutierres Gonçalo
Publication year - 2003
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.200310029
Subject(s) - axiom of choice , mathematics , axiom , axiom independence , urelement , constructive set theory , zermelo–fraenkel set theory , separation axiom , topological space , space (punctuation) , choice function , pure mathematics , discrete mathematics , set (abstract data type) , set theory , computer science , geometry , programming language , operating system
It is known that – assuming the axiom of choice – for subsets A of ℝ the following hold: (a) A is compact iff it is sequentially compact, (b) A is complete iff it is closed in ℝ, (c) ℝ is a sequential space. We will show that these assertions are not provable in the absence of the axiom of choice, and that they are equivalent to each