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1 results

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

  • Marianne Morillon
  • Journal:
    The Journal of Symbolic Logic / Volume 75 / Issue 1 / March 2010
    Published online by Cambridge University Press:
    12 March 2014, pp. 255-268
    Print publication:
    March 2010

  • 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 ℓ 1(I) (resp. such that F ⊆ c 0(I)), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC ℕ) implies that F is compact. This enhances previous results where AC ℕ (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 ℓ 2 (I) is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of is not provable in ZF .