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) = omega . 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) = omegaroot{w(G)} = min {tau >= omega : w(G) <= tau^omega} in case G is connected, and

(iii) tgw(G) = w(G/c(G)) . omegaroot{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)) = omega (and, moreover, w(G) = tgw(G)^+ under the Singular Cardinal Hypothesis).

We also prove that tgw(G) = omega . min {|X|: X subseteq G is a compact Hausdorff space with at most one non-isolated point topologically generating G}.

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