A group topology on the real line that makes its square countably compact but not its cube (Q491800)

The pseudointersection number \(\mathfrak p\) is the smallest cardinality of any \(\mathcal G\in[\omega]^\omega:=\{X\subseteq \omega: |X|=\omega\}\) with the strong intersection property (i.e., every finite subfamily of \(\mathcal G\) has infinite intersection) but with no pseudointersection (i.e., there is no infinite set that is almost contained in every member of \(\mathcal G\)). The authors prove that, under the assumption \(\mathfrak p=\mathfrak c\), the additive group of the real line \(\mathbb R\) admits a Hausdorff group topology \(\tau\) such that \((\mathbb R,\tau)^2\) is countably compact while \((\mathbb R,\tau)^3\) is not countably compact. To this aim, noticing that \(\mathbb R\) is isomorphic to \(H:=\mathbb Q^{(\mathfrak c\times 2)}\), they find a suitable injective group homomorphism \(\Phi:H\to \mathbb T^{\mathfrak c}\), so that \(\Phi(H)\) endowed with the topology inherited from \(\mathbb T^{\mathfrak c}\) has the required property that \(\Phi(H)^2\) is countably compact while \(\Phi(H)^3\) is not countably compact.