2024/07/28 by Melnikov, Alexander G., Nies, Andre
#FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Logic (math.LO)
paper · doi:10.48550/arxiv.2407.19440
Given a computably locally compact Polish space M, we show that its 1-point compactification M^* is computably compact. Then, for a computably locally compact group G, we show that the Chabauty space \mathcal S(G) of closed subgroups of G has a canonical effectively-closed (i.e., Π01) presentation as a subspace of the hyperspace \mathcal K(G^*) of closed sets of G^*. We construct a computable discrete abelian group H such that \mathcal S(H) is not computably closed in \mathcal K(H^*); in fact, the only computable points of \mathcal S(H) are the trivial group and H itself, while \mathcal S(H) is uncountable. In the case that a computably locally compact group G is also totally disconnected, we provide a further algorithmic characterization of \mathcal S(G) in terms of the countable meet groupoid of G introduced recently by the authors (arXiv: 2204.09878). We apply our results and techniques to show that the index set of the computable locally compact abelian groups that contain a closed subgroup isomorphic to (ℝ,+) is arithmetical.