A group topology on the real line that makes its square countably compact but not its cube

Abstract

Under p = c, we show that it is possible to endow the additive group of the real line with a Hausdorff group topology that makes its square countably compact but not its cube.

Countably compact group topologies on torsion-free Abelian groups

“…We prove (#). All numbered references in this proof are from the paper A. H. Tomita, 2015. First, notice that if ([ℎ 𝑖 ]  ∶ 𝑖 < 𝑚) is a ℚ-linearly independent family and the group generated by it does not contain nonzero constant classes, the it satisfies the conclusion of Lemma 4.1. Then, following the proof of Lemma 7.1, using the 𝑓 's as the ℎ's themselves, we see that the functions ℎ 𝑖 for 𝑖 < 𝑚 are integer combinations of the stack 1 𝑁  that was constructed.…”

“…In A. C. Boero and A. H. Tomita, 2010, such a topology was constructed using c selective ultrafilters. In A. Boero et al, 2015, it is shown that there is a group topology without non-trivial convergent sequences such that ℝ 2 is countably compact, a first step to make larger powers of ℝ countably compact.…”