Corollary 1

Tag 02MM, permanent. Cite it as [Bourbaki, Tag 02MM].

Théories spectrales, chapter II, § 2. Classification

Suppose G compact. Then the following conditions are equivalent:

(i) The group G is connected;

(ii) The group $\widehat{G}$ is without torsion;

(iii) The group G is divisible.

This follows from Prop. 4 and Cor. 7 of II, p. 229 since $B_G= G$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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