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Notions of compactness for special subsets of ℝI and some weak forms of the axiom of choice

  • Marianne Morillon (a1)

We work in set-theory without choice ZF. A set is countable if it is finite or equipotent with ℕ. Given a closed subset F of [0, 1]I which is a bounded subset of 1(I) (resp. such that Fc0(I)), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC) implies that F is compact. This enhances previous results where AC (resp. the axiom of Dependent Choices) was required. If I is linearly orderable (for example I = ℝ), then, in ZF, the closed unit ball of the Hilbert space 2 (I) is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of is not provable in ZF.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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