Weight of closed subsets topologically generating a compact group

摘要

Let G be a compact Hausdorff group. A subspace X of G topologically generates G if G is the smallest closed subgroup of G containing X. Define tgω(G) = ω·min{ω(X) : X is closed in G and topologically generates G}, where ω(X) is the weight of X, i.e., the smallest size of a base for the topology of X. We prove that: (i) tgω(G) = ω(G) if G is totally disconnected, (ii) tgω(G) = (w(G)){sup}(1/ω) = min{τ ≥ ω : w(G) ≤ τ{sup}ω} in case G is connected, and (iii) tgω(G) = w(G/c(G)) · ω(c(G))){sup}(1/ω), where c(G) is the connected component of G. If G is connected, then either tgω(G) = ω(G), or cf(tgω(G)) = ω (and, moreover, w(G) = tgω((G)){sup}+ under the Singular Cardinal Hypothesis). We also prove that tgω(G) = ω· min{|X| : X {is contained in} G is a compact Hausdorff space with at most one non-isolated point topologically generating G}.