Weight of closed subsets topologically generating a compact group

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 tgw ( G ) = ω · min{ w ( X ): X is closed in G and topologically generates G }, where w ( X ) is the weight of X , i.e., the smallest size of a base for the topology of X . We prove that: (i) tgw ( G ) = w ( G ) if G is totally disconnected, (ii) tgw ( G ) = $ \root \omega \of {w(G)} $ = min{ τ ≥ ω : w ( G ) ≤ τω } in case G is connected, and (iii) tgw ( G ) = w ( G / c ( G )) · $ \root \omega \of {w(c(G))} $ , where c ( G ) is the connected component of G . If G is connected, then either tgw ( G ) = w ( G ), or cf( tgw ( G )) = ω (and, moreover, w ( G ) = tgw ( G ) + under the Singular Cardinal Hypothesis). We also prove that tgw ( G ) = ω · min{| X |: X ⊆ G is a compact Hausdorff space with at most one non‐isolated point topologically generating G }. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)